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OPTIMA M athematicalOptimizationSocietyNewsletter 86 PhilippeL.Toint MOSChair'sColumn July1,2011. Forthefewsorryspiritswho,misguidedbythegene ralstateoftheworld,thoughtthatmathematicaloptimizationwas intheoreticaldeclineortooremotefromapplicationsinthereal world,theSIOPTConferenceonOptimizationinDarmstadtinMay wasarealandvividcounterexample.Indeedthisverywellattended meeting(600+participants)wasaresoundingsuccessintermsof interestandqualityofthetalks.Ashasbeensofarthecaseinthis seriesofmeetings,thefocuswasmainlyoncontinuousproblems: inparticular,problemsarisingfromcontinuousmechanics,uidsand controlwereprominent,showingtheveryhealthystateofnotonly optimizationinthosedomains,butalsotheGermanindustry'sinterestinoptimizationingeneral.Thesignicantpresenceofdiscrete optimizationwasalsonoticeable,withseveralinterestingsessions andplenarytalksinthisarea.Asoptimizers,allweredelightedthat, oncemore,ahighqualityconferencehasbeenorganizedbySIAM inadditiontothemajoreventsorganizedbytheMathematicalOptimizationSociety. Ifmathematicaloptimizationisturningtodayintooneofthemajor branchesinappliedmathematics,thisisduenotonlytoourpresent effortsasscientists,butalsotothoseofthefoundingfathersofour researchdomain.Oneofthem,CharlesBroyden(theBinBFGS) unfortunatelypassedawayonFriday20thMay,attheageof78(see theobituarypublishedinthisissueonpage10).Hismemorywill staywithusforlong,andhisworkwillundoubtedlycontinueto inspire. Thebeginningof2011wasalsothetimetostartthinkingabout thevariousprizessponsoredbyMOS,whichwillbeawardedinthe InternationalMathematicalProgrammingSymposium inBerlininAugust2012.ItmaybeusefultorecallthattheMOScurrentlyawards vescienticprizesandanamedlectureship.ThesearetheDantzig, Lagrange,Beal-Orchard-Hays,FulkersonandTuckerprizes,andthe PaulTsengLectureship,whosemorecompletedescription,scope andpastwinnerscanbefoundontheMOSWebsite( http://www. m athprog.org ).Therespectivecommitteeshavenowbeenestabl ishedfordecidingtowhomthesedistinctionsmustbegiven,and Iwouldliketotakethisopportunitytothankallofourcolleagues whokindlyacceptedtoserveonthesecommittees.Iwouldalsolike tocallonallmemberstothinkaboutproposinghighqualitysubmissionsfortheseprizes.Iamcertainthattheirscienticvaluecanonly beenhancedbyfriendlycompetitionbetweenhighqualitysubmissions.Iampersonallylookingforwardtomeetingyouallatthethe awardceremonyduringtheopeningsessionoftheBerlinISMP. Thisisalsothetimetostartlookingatpossiblesitesthatwillhost ISMPin2015.Thecallforproposalsubmissioncanbefoundinthis issueonpage12.Andasalways,donotforgettorenewyourMOS membership. Meanwhile,enjoythesummer(forthemajorityofusinthenorthernhemisphere)andletuskeeptheabundanceandqualityofour scienticactivitiesatthepresentvibrantlevel. NotefromtheEditors Thestablesetprobleminclaw-freegraphsisthemaintopicof this issueofOptima.Muchofthetremendousprogressthathasrecently beenobtainedonthisgeneralizationofthematchingproblemis duetoworkoftheauthorsGianpaoloOriolo,GautierStauffer,and PaoloVenturaofthearticleyou'llndbelowandtheirco-workers. Inthediscussioncolumn,ManfredPadbergshareswithushismemoriesofthehistoricalcontextinwhichtheinterestinthestableset probleminclaw-freegraphsaroseandofhowittraveledtoItaly. KatyaScheinberg,Editor SamBurer,Co-Editor VolkerKaibel,Co-Editor ContentsofIssue86/July2011 1 PhilippeL.Toint, MOSChair'sColumn 1NotefromtheEditors 1GianpaoloOriolo,GautierStaufferandPaoloVentura, StableSetsinClaw-FreeGraphs:RecentAchievementsand FutureChallenges 8ManfredW.Padberg, NodePackingsinGraphsandClawFree Graphs 10OlegBurdakov,JohnDennis,andJorgeMor, CharlesG.Broyden,19332011 10CallforNominationsforthe2012Beale-Orchard-HaysPrize 11ISMP2012inBerlin 12Imprint GianpaoloOriolo,GautierStaufferandPaoloVentura StableSetsinClaw-FreeGraphs:Recent AchievementsandFutureChallenges 1Introduction Astablesetinagraph G V;E isasetofverticesthatarepairwise non-adjacent.When G istheintersectiongraphoftheedgesofa graph H twoedgesintersectiftheyshareanendpointastable setin G correspondstoamatchingin H (andvice-versa).Hence whilethestablesetproblemishardingeneral,thespecialcaseof linegraphsthefamilyofallsuchintersectiongraphscanbehandledinpolynomialtimethroughmatching. Matchingisaclassicproblemincombinatorialoptimizationandit exhibitssomeremarkableproperties.Manyofthosepropertieshave

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2 OPTIMA86 beenextendedandledtoverypowerfultoolsandtheorieslike for instancematroidintersectionordelta-matroids.Inordertoextend thematchingtheorytothestablesetsetting,itappearsthattwo fundamentalpropertiesofmatchingarecrucial: theaugmentingpath property andthe intersectionproperty Petersenobservedin1891(andBergeprovedin1952)thatthe symmetricdifferenceoftwomatchingsismadeofalternatingpaths andevencycles.Inparticular,amatching M isofmaximumcardinality inagraph G ifandonlyiftheredoesnotexistanyaugmentingpath in G withrespectto M .Moreover,astwomatchingsareadjacenton thematchingpolytope MATCHG theconvexhullofallincidence vectorsofmatchingsinagraph G ifandonlyiftheyhaveaconnectedsymmetricdifference,onecaneasilyshowthat MATCHG hastheintersectionproperty: MATCHG \f x : P e 2 E x e k g is integralforeveryinteger k Interestinglythosepropertiesextendtothestablesetsettingbeyondlinegraphs(alternatingpathsandcyclesbeingdenedinterms ofverticeshere):theyarealsovalidforstablesetsin claw-freegraphs agraphisclaw-freeifnovertexhasastablesetofsizethreein itsneighborhood.ThiswasobservedbyBergein1973forthesymmetricdifferenceofstablesetsinclaw-freegraphsandbyCalvilloin 1979fortheintersectionproperty.Remarkably,BergeandCalvillo alsoprovedtheconverse,i.e.,aclassofgraphsexhibitsoneorthe otherofthosepropertiesforthestablesetproblemifandonlyif itisasubclassofclaw-freegraphs.Hence,withrespecttostable sets,claw-freegraphsappeartobetherightframeworktoextend theaforementionedpropertiesofmatching.Theproblemofnding amaximumweightedstablesetinclaw-freegraphshasbeenthereforeinvestigatedbyseveralpeople,anditstheoryhasbeendevelopingformorethan40years.Thelast10yearshavebeenparticularly productive,mainlyduetonewapproachesthatexploitresultsfrom structuralgraphtheory.Thepurposeofthispaperistohelptheinterestedresearcherstonavigatethroughthevariousresultsinthis eldandinparticulartoshedlightonthelatestachievementsand thecurrentopenquestions. Forthesakeofshortness,sometheoremsmightbeslightlyimprecise. Inthiscase,areferenceisgiven,andthereadershouldrelyonthat.Also weoftendenoteby VG and EG thevertexsetandtheedgesetofa graph G 2StableSetsinClaw-FreeGraphs:SomeClassical R esults Inthissectionwesurveyafewclassicalresultsontheproble m.We rstdealwithsomealgorithmicresults,andthenmovetosome polyhedralquestions. 2.1AlgorithmsfortheMaximumWeightedStableSetProblem Givenaclaw-freegraph G V;E andaweightfunction w : V ,R amaximumweightedstableset( MWSS )canbefoundinpolynomial time.Wedenoteby G thecardinalityofsuchastablesetwhen w istheallonesvector; G isalsocalledthe stabilitynumber of G Atthepresenttime,thereareseveralalgorithmsfortheproblem, andwemayrecognizethreedifferentmainapproaches.Arstclass ofalgorithmsdealswithaugmentingpathstechniques,andthealgorithmsbyMinty[ 35 ]andSbihi[ 50 ],respectively,fortheweighted a ndtheunweightedcase,followthisapproach.Infact,aswealready discussed,Berge'saugmentingpaththeoremformatchingextends tostablesetsinclaw-freegraphs(apath P is augmenting withrespecttoastableset S if VP n S [ S n VP isastablesetof size j S j +1): Theorem1 ([ 4 ]) Astableset S ismaximumforaclaw-freegraph G ifandonlyiftherearenopathsthatareaugmentingwithrespectto S Sbihi'salgorithmbuildsuponthistheoremwhileMinty'sbuilds uponacuteextensiontotheweightedcase(givenanaugmenting path P withrespecttoastableset S ,theweightofthispathisgiven by wVP n S )Tj /T1_3 8.234 Tf 8.4 0 Td [(wVP \ S ): Theorem2 ([ 35 ]) Let S bea MWSS ofsize k ,andlet P beanaugmentingpathofmaximumweightwithrespectto S .Then S n VP [ VP n S isa MWSS ofsize k 1 Minty'sideaistodetectthosemaximumweightaugmentingpaths andproceedwithatmost j V j augmentations.Giventwoexposed vertices u;v of V n S ,i.e.,theyarebothadjacenttoasinglevertex of S ,Minty'scrucialideaisthatofreducingtheproblemofnding an u )Tj /T1_3 8.234 Tf 8.4 0 Td (v augmentingpathwithmaximumweighttotheproblemof ndingamatchingwithmaximumweightinanauxiliarygraph H .The constructionof H isratherintricate.Wesimplymentionherethat thisgraphhas O j V j vertices.Hencethewholealgorithmrequires thesolutionof O j V j 3 weightedmatchingproblemsinanauxiliary graphwith O j V j vertices. In2001thealgorithmofMintywasslightlyrevisedbyNakamuraandTamura[ 36 ],astheyrealizedthat,intheweightedcase, t healgorithmcouldfailforsomespecialcongurations.Subsequently,Schrijver[ 51 ],elaboratingonMinty'salgorithm,proposed a nelegantalternativeusingaslightlydifferentedge-weightedauxiliarygraph H .Thealgorithmcanbeimplementedtorunintime O j V j 5 log j V jj V j 4 j E j intheweightedcaseandintime O j V j 5 in theunweightedone(however,Sbihiclaimedthatheralgorithm,for theunweightedcase,canbeimplementedtorunintime O j V j 3 Anentirelydifferentapproach,basedonreductiontechniques, wastakenbyLovszandPlummer[ 34 ],forsolvingtheproblemin t heunweightedcase.Thecrucialideahereisthatofperforminga seriesofgraphreductionsthatpreservethestabilitynumber,asto endupwithalinegraph,whereonehastosolvea single matching problem.Theresultingalgorithmisveryelegant,muchlessintricate thanthepreviousalgorithms,and,asLovszandPlummerpointout, withsomecareitcanbeimplementedastorunin O j V j 4 .Unfortunately,inspiteofsomeefforts,itisnotclearhowtoextendthis algorithmtotheweightedcase. However,recentlyNobiliandSassano[ 39 ]wereabletocombine i deasfromboththealgorithmofMintyandthatofLovszandPlummertoprovideanewalgorithmfortheweightedcasethatrunsin O j V j 4 log j V j -time.Ifwecompare(veryroughly!)theiralgorithm withMinty'salgorithm,weseethat,ononehandNobiliandSassanoareabletoreducethenumberofmatchingproblemsthathave tobesolvedto O j V j 2 ,whileontheothertheyareabletosolve eachoftheseproblemin O j V j 2 log j V j -time,thankstoaweighted reduction,inspiredfromthatofLovszandPlummer. Alattersolutionapproachtothe MWSS probleminclaw-free graphsisbasedondecompositiontechniquesandhasbeentaken byOriolo,PietropaoliandStauffer[ 41 ]rst,andbyFaenza,Oriolo, a ndStauffer[ 19 ]later.Thelatteralgorithmcanbeimplementedto r unintime O j V j j V j log j V jj E j .Wepostponethediscussion aboutthesealgorithmstoSection4.1,asitisrstconvenienttodeal withsomestructuraldecompositionresultsforclaw-freegraphs. 2.2StableSetsinClaw-FreeGraphs:PolyhedralIssues Thestablesetpolytope S TABG ofagraph GV;E istheconvex hullofthecharacteristicvectorsofstablesetsin G ,i.e., STABG conv f x 2f 0 ; 1 g j V j j x u x v 1 ; 8f u;v g2 E g .Sincetheseminal paperbyPadberg[ 43 ],thispolytopehasbeencarefullyinvestigated b yseveralauthors(seee.g.[ 12 37 38 ]). B ecausethe MWSS probleminclaw-freegraphscanbesolvedin polynomialtime,exactseparationoverthispolytopealsocanbe doneinpolynomialtime[ 28 ],andhencethestablesetpolytopeof

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July2011 3 claw-freegraphsissomewhatundercontrol.Howevernocom pletelineardescriptionisknownatthetimeofwriting,despitethe factthattheproblemisofciallyopenformorethanaquarterof acentury[ 29 ]:inspiteofconsiderableefforts,nodecentsystem o finequalitiesdescribingSTAB(G)forclaw-freegraphsisknown. Suchadescriptionwouldpossiblyresultinaniceminmaxcharacterizationoftheproblem. Aneatdescriptionisathandforthestablesetpolytope oflinegraphs.Indeed,Edmonds[ 15 ]provedthatthematchi ngpolytope MATCHH ofagraph HV;E i.e., conv f x 2 f 0 ; 1 g j E j j P e 2 v x e 1 ; 8 v 2 V g canbedescribedbynonnegativityinequalities,degreeinequalities(asusual,wedenoteby v thesetofedgesincidenttoanode v ),andoddsetsinequalities,where,foranoddset S V ,wedenoteby ES thesetof edgesbetweenverticesof S Theorem3 ([ 15 ]) Thematchingpolytopeofagraph HV;E can becharacterizedas MATCHH f x 2 R j E j j x 0; P e 2 v x e 1 ; 8 v 2 V ; P e 2 ES x e b j S j 2 c g ,foreveryoddset S V g : Butsince MATCHH STABLH ,where LH denotesthe linegraphof H ,itfollowsthatthestablesetpolytopeoflinegraphs canbedescribedbynon-negativityinequalities,cliqueinequalities and Edmonds'inequalities ,thecounterpartofoddsetinequalitiesin thestablesetsetting.Moreformally: Denition2.1. Foragraph GV;E andanoddsetofcliques K ,let V 2 K bethesetofverticescoveredbyatleast2cliquesof K .The Edmonds'inequality associatedwith K is: P v 2 V 2 K x v b jKj 2 c F romDenition2.1itfollowsthatEdmonds'inequalitiesarederivedasChvtal-Gomorycutsfromthe cliquerelaxation ofthestable setpolytope QSTABG : f x 2 R j V j : x 0; xK 1 ,forevery clique K of G g [ 17 ],i.e.,theycanbeobtainedbyrsttakinganonn egativecombinationoftheinequalitiesdescribing QSTABG ,and thenroundingdowntherighthandsideofthecombination. Lemma2.1 ([ 15 ]) Foralinegraph G ,non-negativityinequalities, cliqueinequalitiesandEdmonds'inequalitiesareenoughtodescribethe stablesetpolytope. Unfortunately,Lemma2.1doesnotholdtrueforclaw-free graphs.Infact,considera5-wheel,i.e.,agraphwithvertexset f w;v 1 ;v 2 ;v 3 ;v 4 ;v 5 g andedgeset f w;v i ;v i ;v i 1 forall i 1 ;::; 5 g with v 6 v 1 ,thenthe 5-wheelinequality P 5 i 1 x v i 2 x w 2 isafacetofitsstablesetpolytope.Thisshowsthat non-rank inequalitiesareneededinordertodenethestablesetpolytope ofclaw-freegraphs.Aninequalityis rank ifitonlyinvolves{0,1}valuedcoefcientsinthelefthandside,i.e.,ifitisoftheform P v 2 S x v GS for S V In1978Maurras,inferringthat5-wheelsand,moregenerally, odd-antiwheels(i.e.,agraphmadeofavertextotallyjoinedtothe complementofanoddhole)weretheproblem,introducedtheclass of quasi-linegraphs ,i.e.,claw-freegraphswithoutodd-antiwheels.He alsoconjecturedthatforquasi-linegraphsallfacetsof STABG are rank.BuildinguponMaurras'conjecture,Sbihiconjecturedthatfor claw-freegraphsallfacetsof STABG haveonly{0,1,2}-valuedcoefcients.BothconjectureswereprovenfalsebyGilesandTrotter [ 27 ]in1981(seeFigure2and1,respectively).Weknownowthat f orclaw-freegraphswithstabilitynumber3thereexistfacetswith arbitrarilymanycoefcients[ 46 ]andthatforanyinteger a t hereexistquasi-linegraphswhosestablesetpolytopesinvolvefacetswith coefcient a and a 1 [ 27 33 ].WhileMaurras'conjecturewas w rong,hisintuitionontherelevanceoftheclassofquasi-linegraphs wascorrect.Indeed,incontrastwithgeneralclaw-freegraphs, 1 4 3 2 5 7 6 9 8 1 0 Figure1. Thecomplementofaclaw-freegraph G .Thegraph G inducesthe facet: 2 x 1 2 x 2 2 x 3 2 x 4 2 x 5 x 6 x 7 3 x 8 3 x 9 3 x 10 4 .Note that G isnotquasi-lineandthat G 3 .(ThepictureisacourtesyofTristram Bogart,AnnieRaymondandRekhaThomas.) thenatureoftheinequalitiesneededforquasi-linegraphswas graspedrstbyBenRebea[ 49 ]andlaterbyOriolo[ 40 ]whonamed a conjectureafterhim:theBenRebeaconjecture. TheBenRebeaConjecture1 ( Oriolo[ 40 ]) Foraquasi-linegraph G ,non-negativityinequalities,cliqueinequalitiesandcliquefamilyinequalitiesareenoughtodescribe STABG Denition2.2. Givenagraph G ,afamilyofcliques K andaninteger p 2 ,dene V p K and V p )Tj /T1_4 6.228 Tf 4.92 0 Td (1 K asthesetofverticescoveredby atleast p cliquesandexactly p )Tj /T1_4 8.234 Tf 8.88 0 Td (1 cliques,respectively.Thefollowing inequalityisvalidfor STABG andiscalledthe cliquefamilyinequality associatedwith K and p : P v 2 V p K x v p )Tj /T1_2 6.228 Tf 4.92 0 Td (r )Tj /T1_4 6.228 Tf 4.92 0 Td (1 p )Tj /T1_2 6.228 Tf 4.92 0 Td (r P v 2 V p )Tj /T1_4 4.638 Tf 3.72 0 Td (1 K x v b j Kj p c where r jKj mod p CliquefamilyinequalitiesgeneralizeEdmonds'inequalities, andtheirvaliditycaneasilybederivedbythedisjunction P v 2 V p K [ V p )Tj /T1_4 4.638 Tf 3.72 0 Td (1 K x v b jKj p c P v 2 V p K [ V p )Tj /T1_4 4.638 Tf 3.72 0 Td (1 K x v b jKj p c 1 appliedto QSTABG TheBenRebeaconjecturesuggestedthatthestablesetpolytope ofquasi-linegraphshasaneatdescription.Asforclaw-freegraphs, in1991GalluccioandSassano[ 26 ]providedanelegantcharacteriz ationof rankminimalfacets ,i.e.,rankfacetsthatareminimalwith respecttoliftingandcompletejoinoperations[ 11 43 ]. W eclosethissectionbyillustratingtheresultofCalvillo[ 5 ]that w ementionedbefore.Calvilloprovedthefollowingnicepropertyof thestablesetpolytopeofclaw-freegraphs.Apolytope P R n has the intersectionproperty if P \f x 2 R n : P n i 1 x i k g isintegralfor allinteger k Theorem4 ([ 5 ]) STABG hastheintersectionpropertyifandonly if G isaclaw-freegraph. Figure2. Aquasi-linegraphinducingthefacet P v 2 x v 2 P v 2 x v 6 .On theright,thecliquesinvolvedinthederivationoftheinequalityasacliquefamily inequality.

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4 OPTIMA86 3 ABreakthrough:DecompositionofClaw-Freeand Quasi-LineGraphs Inalongseriesofpaper,ChudnovskyandSeymour(seee.g. [ 8 9 10 6 ])elucidatethestructureofclaw-freegraphsanddeneadecompositionresultforthem.Forthispurpose,theyhave introducedanewcompositionoperation. 3.1TheCompositionofStrips Inordertobettergraspthisoperation,itisconvenienttor stdeal withanalgorithmicprocedurethatcanbeusedtobuildlinegraphs. Therationaleofthislatteroperationisthefollowing.Givenagraph G ,eachvertexin G canbeassociatedwithacliqueinthelinegraph H LG (alledgesincidenttothisvertexarepairwiseadjacentin H ).Ifwelet F denotethefamilyofcliquesof H thatareassociated withverticesof G ,weobservethat F hasthefollowingproperties: (i)everyedgeof H iscoveredbysomecliqueof F ;(ii)everyvertex of H iscoveredbyexactlytwocliquesof F Supposenowthatwearegivena(general)graph H .Wecalla family F ofcliquesof H a Krauszfamily ifitsatisestheaboveproperties.Krausz[ 32 ]provedthefollowing: T heorem5 ([ 32 ]) Agraphisthelinegraphofamulti-graphifand onlyifitadmitsaKrauszfamily. Thistheoremgivesanalgorithmicproceduretobuildlinegraphs. Thisprocedurerequiresasinputasetofvertices V andapartition P P 1 ;:::;P q ofthemulti-set V [ V .Itthenassociates tothepair V; P thegraph G withvertexset V andedgeset E : ff u;v g : u v andboth u;v 2 P i ,forsome 1 i q g ChudnovskyandSeymourgeneralizedtheaboveconstruction,essentiallybyreplacing vertices with strips .Weborrow(butslightly change)somedenitionsoftheirs. Denition3.1. A strip G; A isagraph G (notnecessarilyconnected)withamulti-family A ofeitheroneortwodesignatednon-empty cliques(possiblyidentical)of G .Thecliquesin A arecalledthe extremities ofthestrip. Let Hf G i ; A i i 1 ;:::;k g beafamilyofvertexdisjoint strips.Let A H denotethemultifamilyoftheextremitiesofthose strips,i.e., A H S i 1 ::k A i ,andlet P P 1 ;P 2 ;:::;P q bethe classesofapartitionof A H .Weassociatetothepair H ; P the graph G thatismadeofthedisjointunionofthegraphs G 1 ;:::G k withadditionaledges E : ff u;v g : u v and u and v belongto differentextremitiesinasameclass P i ,forsome 1 i q g G is calledthecompositionofthestrips H withrespecttopartition P Notethat,forlinegraphs,thiscompositionreducestotheabove construction,assoonaseachgraph G i ismadeofasinglevertex v i andthecorrespondingstripis f v i g ; ff v i g ; f v i gg Eventhoughtheoperationofcompositionofstripsbuildsgraphs thatareingeneralnon-line,suchgraphsindeedinheritalinestructurefromitssimilaritywithKrauszcomposition.Saythatastrip H G; A is line if G admitsaKrauszfamily K with AK .Then, assoonasall strips are line ,thecompositionisalinegraph.The proofofthisfactisstraightforward.Wewillmakeheavyuseofthis factinthefollowing. Lemma3.1. Let G bethecompositionofafamilyof line strips H i G i ; A i ;i 1 ;:::;k withrespecttoapartition P .Then G is alinegraph. 3.2DecompositionResultsforClaw-FreeandQuasi-LineGrap hs In[ 8 ]ChudnovskyandSeymouroverviewaseriesofpapersinwhich t heyproveastructuretheoryforclaw-freegraphs.Thetheoryistoo complextodescribeindetailhere,sowejustoutlinetwooftheir results. Theorem6 ([ 10 ]) Let GV;E beaconnectedclaw-freegraph.Then oneofthefollowingsholds:i) G 3 and G belongstoasmallsetof basicgraphs;ii) G isafuzzycircularintervalgraph;iii) G isthecompositionofstrips,thatareeitherfuzzylinearintervalstripsortheybelong tooneofasmallnumberoffamilyofstrips,allwithstabilitynumberat most 3 Circularintervalgraphs aredenedbyasetofvertices,acircleand asetofarcs.Verticesaremappedtothecircleandtwoverticesare adjacentifandonlyiftheyarecoveredbyanarc.Thosegraphsare alsoknownaspropercirculararcgraphs.Linearintervalgraphsare constructedinthesamewayascircularintervalgraphs,butonaline ratherthanonacircle.Inalinearintervalstrip G; A G isalinear intervalgraphandthecliquesin A aremadeofcontiguousvertices attheendofthelinesegment. Fuzzycircular/linearintervalgraphs are aslightgeneralizationofcircular/linearintervalgraphs. Theresultsconsiderablysimplifyforthesubclassofquasi-line graphs. Theorem7 ([ 8 ]) Let GV;E beaconnectedquasi-linegraph.One ofthefollowingsholds: G isafuzzycircularintervalgraph; G isthecompositionoffuzzylinearintervalstrips. Wepointoutthatwhilethetwoaboveresultsarenotalgorithmic,lighterversionsofthosehavebeenrecentlyalgorithmizedby Hermelin,Mnich,vanLeeuwenandWoeginger[ 30 ]. A differentalgorithmicdecompositiontheoremforclaw-free graphswasrecentlygivenbyFaenza,OrioloandStauffer.Froma structuralpointofview,thisresultismuchweakerthanTheorem6; however,itisparticularlyusefulwhendealingwiththe MWSS problem,aswediscussinSection4.1. Theorem8 ([ 19 ]) Let GV;E beaclaw-freegraph.Intime O j V jj E j ,onecanndoutwhether G 3 ,or G isalmostnearlydistancesimplicial,or G isthecompositionof O j V j stripsthataredistance simplicialstripsorstripswithstabilitynumberatmost3andcontaininga 5 -wheel(andprovidethedecomposition). Wedenoteby NS ,with S V ,thesetofnodesof V n S that areadjacenttosomenodein S (wealsousethenotation Nv for N f v g ).Aconnectedgraph G is distancesimplicialwithrespectto aclique K if,forevery j N j K 1 ,i.e.,eachneighborhood N j K of K isaclique;ifthereexistsavertex v suchthat G is distancesimplicialwithrespectto f v g ,wesimplysaythat G is distancesimplicial .A distancesimplicialstrip isastrip G; A ,suchthat G isdistancesimplicialwithrespecttoeachcliquein A .Agraphis nearly distancesimplicialif,foreach v 2 V G n Nv [f v g isdistancesimplicial. Almost nearlydistancesimplicialgraphsareaslight generalization.When G isquasi-line,Theorem8readsasfollows: Theorem9 ([ 19 ]) Let GV;E beaquasi-linegraph.Intime O j V jj E j ,onecanndoutwhether G isthecompositionof O j V j distancesimplicialstrips,or G isalmostnearlydistancesimplicial(and providethedecomposition). 4FollowingtheBreakthrough:StableSetsinClaw-Free G raphsRevisited 4.1FasterAlgorithmforClaw-FreeGraphs Supposethatweareinterestedinsolvingthe M WSS problemon agraph G thatisthecompositionofsomestrips H 1 ;:::;H k and that,inparticular,weareabletosolvethesameproblemoneach strip.BuildinguponLemma3.1wemayreducetheformerproblem toamatchingproblem.Thisgoesasfollows:wereplaceeachstrip H i ;i 1 ;:::;k withsuitable,simple,linestrips H i ; i 1 ;:::;k ,and

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July2011 5 considerthegraph G obtainedbysubstituting H i w ith H i i nthe composition.FollowingLemma3.1, G isalinegraph,andtherefore a M WSS of G canbefoundbysolvingamatchingproblem.Finally, f roma MWSS of G wethenrecovera M WSS of G .Wehaveinfact: Theorem10 ([ 41 ]) Themaximumweightedstablesetproblemona graph G thatisthecompositionofsomestrips G 1 ; A 1 ;:::;G k ; A k canbesolvedin O j VG j 2 log j VG j P i 1 ;:::;k p i j VG i j -time, ifeach G i belongstosomeclassofgraphs,wherethesameproblemcan besolvedintime Op i j VG i j Faenza,OrioloandStauffer[ 19 ]recentlyproposedastrongly p olynomialalgorithmforsolvingthe MWSS probleminaclaw-free graph GV;E thatrunsin O j V j j E jj V j log j V j )Tj /T1_1 9.465 Tf 6.6 0 Td [(time,drasticallyimprovingthepreviousbestknowncomplexitybound.This algorithmbuildsuponTheorem8,and,inthefollowing,wesketch howitdealswiththedifferentcasesarisingfromthattheorem.Let G beaclaw-freegraph.If G 3 ,thena MWSS canbefoundby enumeration.If G isthecompositionofstrips,thentheresultfollowsfromTheorem10,assoonasweobservethatwecannda MWSS inadistancesimplicialstripbydynamicprogramming,followingaconstructionandanalgorithmfromPulleyblankandShepherd [ 48 ]fordistance-claw-freegraphs.Thelatterconstructionca nbe usedalsowhen G isalmostnearlydistancesimplicial.Withoutusing anysophisticateddatastructures,thealgorithmcanbeimplemented astorunin O j V j j E jj V j log j V j )Tj /T1_1 9.465 Tf 6.6 0 Td [(time. 4.2TheStableSetPolytopeofQuasi-LineGraphs Whenstudyingthepolyhedralaspectofacompositionofgraph s,it isstandardtosubstitutesomeofthegraphswithgadgetsandto derivethepolyhedraldescriptionofthecompositionfromthepolyhedraldescriptionsofthecompositionofthesimplergraphs[ 2 3 ]. F orthecompositionofstrips,ChudnovskyandSeymourobserved [ 7 53 ]thatpathsoflengthoneortwoweretheappropriategadgets t oprovethefollowing: Theorem11 ([ 7 53 ]) Thestablesetpolytopeofaquasi-linegraph G thatisnotafuzzycircularintervalgraphcanbecharacterizedbynonnegativityinequalities,cliqueinequalitiesandEdmonds'inequalities. OnonehandthistheoremshowsthattheBenRebeaconjecture holdsforsuchclassofquasi-linegraphs(Edmonds'inequalitiesare particularcliquefamilyinequalities).Ontheother,withthehelpof Theorem7,itshowsthatallnon-rankfacetinducinginequalitiesfor quasi-linegraphsappearinfuzzycircularintervalgraphs. Eisenbrand,Oriolo,StaufferandVentura[ 17 ]wereabletoprov idealineardescriptionof STABG ,when G isafuzzycircularintervalgraph.Asfuzzinesscanbehandledeasily,inthefollowingwe simplydealwithcircularintervalgraphs.Forsuchgraphs,let A be thecliqueincidencematrix,whenonerestrictstocliquesstemming fromtheintervals.Thenthestablesetproblemcanbeformulated as: max f P v 2 V c v x v : Ax 1; x v 2f 0 ; 1 g ; 8 v 2 V g .Whatis crucialisthatthematrix A hastheso-calledcircularoneproperty, i.e.,thereisanorderingofthecolumnssuchthat,oneachrow,the onesappearconsecutively,undertheconventionthattherstcolumnisconsecutivetothelast.Butthenthelinearrelaxation P f x 2 R n j )Tj /T1_2 6.228 Tf 6.72 -3.48 Td (A )Tj /T1_2 6.228 Tf 4.92 0 Td (I x )Tj /T1_4 6.228 Tf 5.28 -3.36 Td (1 0 g issuchthat P k P \f x : P v 2 V x v k g isintegralforanyinteger k (usingtheequation P v 2 V x v k ,the system )Tj /T1_2 6.228 Tf 6.84 -3.48 Td (A )Tj /T1_2 6.228 Tf 4.92 0 Td (I x )Tj /T1_4 6.228 Tf 5.28 -3.48 Td (1 0 canberewrittenasaconsecutiveonesystem andso P k isdescribedbyatotallyunimodularsystem).Thisresult showsthattheonlymissinginequalitiesaredisjunctivecutsofthe form P v 2 V x v k P v 2 V x v k 1 from QSTABG .Careful analysisofthosedisjunctivecutsallowsonetoprovethattheyare cliquefamilyinequalitiesandthereforetheBenRebeaconjecture holdstrue[ 17 ]. T heorem12 ([ 17 ]) Conjecture1holdstrue. 4.3TheStableSetPolytopeofClaw-FreeGraphswith 4 a ndNo CliqueCutsets Galluccio,GentileandVentura[ 22 ]extendedTheorem11todeal w iththestablesetpolytopeofagraphthatisthecomposition of arbitrary strips.Let G bethecompositionofafamilyofstrips H i G i ; A i ;i 1 ;::;k .Stripswithonlyoneextremitycanbe easilyhandledbecauseofaresultofChvtal[ 12 ].Thereforeassume w ithoutlossofgeneralitythateachstriphastwoextremities.We denoteby G i z thegraphobtainedfrom G i byaddinganewnode z with Nz A i 1 [ A i 2 ,andby G i uv thegraphobtainedfrom G i by addingtwonewnodes u and v suchthat Nu A i 1 [f v g and Nv A i 2 [f u g .In[ 22 ]itisprovedthattheinequalitiesneededto d escribe STABG canbeobtainedby(appropriately)replacingthe inequalitiesdening STABG i uv and STABG i z inthestableset polytopeofacertainlinegraph G ,derivedfrom G bysubstituting each H i withalinestrip. Inthefollowing,weapplythisresulttoclaw-freegraphs.Galluccio,GentileandVentura[ 24 25 ]managedtoprovideadescript ionofthestablesetpolytopeofthegraphs G i z and G i uv associatedwiththestrips G i ; A i arisingfromTheorem6:inparticular, theyshowedthatnon-negativityinequalities,rankinequalities,sequentialliftings[ 43 ]of5-wheelinequalitiesandsequentialliftingsof g earedinequalitiesaresufcienttodescribeboth STABG i z and STABG i uv .Wehavetherefore: Theorem13 ([ 25 ]) Thestablesetpolytopeofaclaw-freegraphwith stabilitynumberatleast4,non-fuzzycircularintervalandwithnocliquecutsetisdenedby:non-negativeinequalities,sequentialliftingsofmultiplegearedinequalities,rankinequalitiesandsequentialliftingsof5-wheel inequalities.Inparticular,allinequalitiesare f 0 ; 1 ; 2 g -valued. Werecallthatadescriptionofrankinequalitiesinclaw-free graphsfollowsfromthecharacterizationof rankminimalfacets in [ 26 ].Asfor (multiple)gearedinequalities [ 21 23 ],inclaw-freegraphs t heyare f 0 ; 1 ; 2 g -valuedfacetdeninginequalitiesthatareproducedfromrankinequalities,bysubstitutingoneormultipleedges witha gear ,agraphthatismadeoftwointertwined5-wheels(see Figure3). CombiningTheorem6,Theorem12andTheorem13,wehave: Theorem14 ([ 25 ]) Thestablesetpolytopeofanyclaw-freegraph G withoutaclique-cutsetandsuchthat G 4 isdenedby:nonnegativityinequalities,clique-familyinequalities,rankinequalities,sequentialliftingsof5-wheelinequalities,andsequentialliftingsofmultiplegeared inequalities. x 3 x 2 x 5 x 2 x 7 F igure3. Arank,agearedandamultiplegearedfacetdeninginequality 4.4ExtendedFormulationandSeparationfortheStableSetPo lytope ofClaw-FreeGraphs Faenza,OrioloandStauffer[ 19 ]gaveacharacterizationof S TABG G claw-free,inanextendedspace.Theirresultbuilds uponasuitableextendeddescriptionof STABG foragraph G

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6 OPTIMA86 thatisthecompositionofafamilyofstrips H i G i ; A i ;i 1 ;::;k Infact,Theorem10hasapolyhedralinterpretationinanextended space.WeproceedasinSection4.1,andlet G bethelinegraph t hatarisesbysubstitutingeachstrip H i withasuitablelinestrip H i i nthecomposition.Givenapolytope P f x 2 R n : Ax b g in R n ,wecall f x; P 2 R n R : Ax P b; P 0 g thehomogenizedconeassociatedwith P andthesystem Ax P b; P 0 the homogenizationofthesystem Ax b Theorem15 ([ 18 ]) Anextendedformulationfor STABG is (mainly)givenbythehomogenizationofthe(possiblyextended)lineardescriptionsof STABG i STABG i VG n[ A2A i A and STABG i VG n A forall A 2A i ;and STABG where G is a linegraphwith O k vertices. (Thereadershouldrelyon[ 18 ]tondoutwhatisbehindthe w ordmainlyinthepreviousstatement.)Wenowsketchhowto applythisconstructiontoaclaw-freegraph GV;E .ByTheorem8, weknowthatintime O j V jj E j wecandistinguishif G 3 ,if G isalmostnearlydistancesimplicialorif G isthecompositionof distancesimplicialstripsandstripswithstabilitynumberatmost3. Itiseasytowriteanextendedformulationforagraph G with smallstabilitynumber.Indeed,let x 1 ;:::;x k bealltheextreme pointsof STABG i.e.,allstablesetofsize 0 ; 1 ; 2 or3.Thepolytope Q f x; : x P k i 1 i x i ; 0 ; P k i 1 i 1 g isanextendedformulationof STABG .Nearlydistancesimplicialgraphs aredistanceclaw-freegraphs,aclassofgraphsforwhichPulleyblank andShepherd[ 48 ]gaveacompactextendedformulationbasedon a dynamicprogrammingalgorithmforthe MWSS problem(notethat thisclassalsoincludesgraphsthataredistancesimplicialgraphswith respecttosomeclique).Weareleftwiththecasewhere G isthe compositionofafamilyofstrips H i G i ; A i ;i 1 ;::;k .Inthis case,buildinguponTheorem15,wejustneedtoshowthatweare abletoderiveextendedformulationsforthestablesetpolytopes associatedwiththestrips.Infact,ifeither G i isadistancesimplicial graphwithrespecttosomeclique,or G i 3 ,thenanextended formulationfor STABG i (or STABG i VG n[ A2A i A etc.)followsfromtheabovearguments. Wepointoutthattheresultingextendedformulationissimple (ageneralizationoftheunionofpolytopes[ 1 ])andrequiresonly O n extravariables.Moreover,eventhoughitmighthaveexponentiallymanyEdmonds'inequalities,theyareseparableinpolytime [ 44 ].Thusonecanwriteanexplicitlinearformulationofthepro blemthatcouldalsobeusedasastrongrelaxationforthevariation ofthestablesetprobleminclaw-freegraphwithadditionalsideconstraints.Oneshouldalsoobservethatiftherewouldexistacompact extendeddescriptionofthematchingpolytope,awell-knownopen problem,thenalsothisformulationwouldbecompact. Faenza,OrioloandStauffer[ 18 ]gaveanotherextendedformulat ionof STABG G claw-free,thatisbettersuitedforprojection,as itrequiresonlyoneadditionalvariableperstrip.Theyderivedfrom theprojectionofthisformulationontheoriginalspaceanalternativeviewtoTheorem14and,moreimportant,apolytimeseparation routinefor STABG (intheoriginalspace). Theorem16 ([ 18 ]) Let GV;E beaclaw-freegraph.Itispossibleto separateinpolynomialtimeover STABG usingonlyaseparationroutineformatchingandthesolutionof O j V j compactlinearprograms. 5OpenQuestions 5 .1CompleteLinearDescriptionof STABG intheOriginalSpace Itfollowsfrom[ 12 ]andTheorem14,thatinordertoprovidea l ineardescriptionofthestablesetpolytopeof any claw-freegraph, 2 1 1 4 10 17 13 8 4 16 15 3 12 7 5 9 6 11 Figure4. Thecomplementofaclaw-freegraph G .Thegraph G inducesthe facet: 2 x 14 x 15 x 16 3 x 1 x 2 x 3 x 4 4 x 5 x 6 x 7 x 9 x 12 5 x 8 x 10 x 11 6 x 13 x 17 8 .Notethat G isnotquasi-lineand that G 3 .(ThepictureisacourtesyofTristramBogart,AnnieRaymondand RekhaThomas.) weareleftwithcharacterizingthestablesetpolytopesofclaw-free graphswithstabilitynumberatmost3.Infact,followingTheorem 12wemayrestricttoclaw-free,non-quasi-linegraphswithstability numberatmost3.However,evenasCook[ 14 52 33 ]characteri zedthestablesetpolytopeofanygraph G with G 2 ,itseems thatcharacterizingthestablesetpolytopesofgraphswithstability numberatmost3,isquitechallenging,evenifwerestricttoclawfreenon-quasi-linegraphs.Infact,wealreadypointedoutinSection 2.2thatforclaw-free,butnonquasi-line,graphswithstabilitynumber3thereexistfacetswitharbitrarilymanycoefcients[ 46 ];see F igure4forafacetinducinginequalitywith5differentcoefcients. Thisisnotthecaseforquasi-linegraphs(seeTheorem12)orclawfreegraphswithstabilitynumberatleastfourandnoclique-cutsets (seeTheorem13). PcherandWagler[ 45 ]workedonthisquestion.Whileproviding s omebetterunderstandingofafneindependencefortheremaining difcultfacetsofSSPinclaw-freegraphstheso-called co-spanning foreststructure thisworkleavesthefullcharacterizationintheoriginalspacestillopen.Indeed,PcherandWagler[ 45 ]donotprovide a `proper'constructiontoproduceavalidinequalityassociatedwith agivenstructurebesides,basically,exploitingthepolarofthepolytope. Observethat,incaseonecansolvethequestionabove,the essenceofthecompletelineardescriptionforclaw-freegraphswill besignicantlydifferentfromthatofthestablesetpolytopeofquasilinegraphs,forwhichinequalitiesaredenedalgebraically.This suggeststhat,evenifthecase G 3 wassolved,additionalinsightmightstillbeneededtogetanelegantdescriptionofthestable setpolytopeofclawfreegraphs,ifany.Thenextquestionproposes anotherstandpointontheproblemthatmightleadtoasimplerdescriptionofthepolytope. 5.2Calvillo'sTheoremandtheIntersectionProperty Let STAB k G : conv f x 2f 0 ; 1 g j V j : x u x v 1 ; 8f u;v g2 E and P v 2 V x v k g .Theorem4showsthat STAB k G STABG \f x 2 R j V j j P v 2 V x v k g anditmightsuggestthat thestablesetpolytopeofclaw-freegraphshasanicerinterpretation whenintersectedwiththehyperplanes f x 2 R j V j j P v 2 V x v k g forallinteger k .Thisisindeedthecaseforquasi-linegraphs.In fact,buildinguponTheorem4,Theorem7,Theorem11andsome argumentsfrom[ 17 ],onecanshowthat: L emma5.1. Let GV;E beaquasi-linegraph.Foreveryinteger k STAB k G canbedescribedbynon-negativityinequalities,cliqueinequalities,Edmonds'inequalitiesand P v 2 V x v k .

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July2011 7 Canwehopeforasimilarresultforclaw-freegraphs?Because o f5-wheelstructure,onecaneasilyshowthat,incontrasttoquasilinegraphs,forclaw-freegraphsrankinequalitiesarenotenough todescribe STAB k G ;howeveracompletecharacterizationof STAB k G mightstillbesimple. 5.3MinimalLinearDescriptionfortheStableSetPolytopeof Q uasi-LineGraphs Forthematchingpolytope,EdmondsandPulleyblank[ 16 ]gaveades criptionofthe facets ofthepolytope,givingnecessaryandsufcient conditionsforanoddsetofverticestoinduceafacet.Whilethe BenRebeatheoremgivesalineardescriptionof STABG when G isquasi-line,itdoesnotprovidenecessaryandsufcientconditions forafamilyofcliques K andaninteger p 2 toinduceacliquefamilyinequalitythatisfacetinducing.Thequestionisopenbutthere areafewresultsinthisdirection[ 26 53 54 42 ]. 5.4TheChvatalGomoryRankoftheStableSetPolytopeof Q uasi-LineGraphs Sbihi[ 50 ]reportedthatEdmondsconjecturedin1973thatthestab lesetpolytopeofanyclaw-freegraph G hadChvtal-Gomoryrank (CG-rankinthefollowing)onefrom QSTABG .Thiswasproven falsebyGilesandTrotter[ 27 ],whoprovidedafacet-deningine qualityfor STABG withCG-ranktwo.TheresultwasstrengthenedbyChvtal[ 13 ]whoshowedthatthereexistgraphswith G 2 ,andthereforeclaw-free,withCG-rankunbounded.Interestinglythisconstructiondoesnotextendtoquasi-linegraphs,as buildinguponTheorem12andresultsin[ 14 52 33 ],onemayshow t hattheCG-rankofaquasilinegraph G with G 2 isone. However,facet-deninginequalitieswithCG-ranktwoexistalsofor quasi-linegraphs,asshownbyOriolo[ 40 ].Thisraisesthefollowi ngquestions:istheCG-rankunboundedforquasi-linegraphsoris itbounded?(Actually,despitesomeefforts,wecouldnotproduce forquasi-linegraphsfacet-deninginequalitieswithCG-rankbigger thantwo).WementionthatPcherandWagler[ 47 ]studiedthe C G-rankofgeneralcliquefamilyinequalitiesandgavesomeupper bounds.Unfortunatelytheydonotseemtobetight. 5.5ImprovingtheComplexity Theweightedmatchingprobleminagraph H W;F canbesolved in O j W j j W j log j W jj F j )Tj /T1_1 9.465 Tf 6.6 0 Td [(time[ 20 ].Itfollowsthatwecannd a M WSS inalinegraph GV;E intime O j V j 2 log j V j .FollowingthealgorithmbyFaenza,OrioloandStaufferpresentedinSection4.1a MWSS inaclaw-freegraph GV;E canbefoundintime O j V j j V j log j V jj E j ,i.e.,slightlyworsethanforlinegraphs:can weclosethisgap?Webelievethatthisshouldbedoable,inparticular forquasi-linegraphs.Alsonotethattheabovealgorithmusesonly elementarydatastructures,soonecouldtrytoloweritscomplexity byusingmoresophisticateddatastructures. 5.6AShortProofoftheBenRebeaTheorem OneshouldnotethatTheorem9isquiteclosetoTheorem7.Howe ver,whiletheformertheoremhasarathersimpleanddirectproof, theproofofthelatterreliesonthegeneralstructureofclaw-free graphs.WeaskthereforewhetheritispossibletosharpenTheorem9soastoprovethesamecharacterizationofTheorem7.The questionofhavingadirectproofofTheorem7wasraisedalready byKing[ 31 ].Notethatsuchaproof,togetherwiththeproofof T heorem11andtheproofthattheBenRebeaconjectureholdsfor fuzzycircularintervalgraphsin[ 17 ],wouldprovideashortproof o fTheorem12. 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[39]P.NobiliandA.Sassano.Areductionalgorithmfortheweightedstablesetprobleminclaw-freegraphs.In Proceedingsofthe10-thCologneTwenteWorkshop ,pages223226,2011. [40]G.Oriolo.Cliquefamilyinequalitiesforthestablesetpolytopefor quasi-linegraphs. DiscreteAppliedMathematics,132(3):185201,2003. [41]G.Oriolo,U.Pietropaoli,andG.Stauffer.Anewalgorithmforthe maximumweightedstablesetprobleminclaw-freegraphs.In Proceedingsofthe12thIPCOConference ,volume5035,pages7796,2008. [42]G.OrioloandG.Stauffer.Clique-CirculantfortheStableSetPolytope ofQuasi-lineGraphs. MathematicalProgramming,SeriesA ,2008. [43]M.Padberg.Onthefacialstructureofsetpackingpolyhedra. MathematicalProgramming ,5:199215,1973. [44]M.W.PadbergandM.R.Rao.Oddminimumcut-setsand b -matchings. MathematicsofOperationsResearch ,7:6780,1982. [45]A.Pcher,U.Pietropaoli,andA.Wagler.Personalcommunication, 2006. [46]A.PcherandA.K.Wagler.Onfacetsofstablesetpolytopesofclawfreegraphswithstabilitynumber3. DiscreteMathematics,310(3):493 498,2010. [47]A.PcherandA.K.Wagler.Anoteonthechv tal-rankofcliquefamily inequalities. RAIROOperationsResearch ,41:289294,2007. [48]W.R.PulleyblankandB.Shepherd.Formulationsofthestablesetpolytope.InG.RinaldiandL.Wolsey,editors, ProceedingsThirdIPCOConference ,pages267279,1993. [49]A.BenRebea. tudedesstablesdanslesgraphesquasi-adjoints.PhD thesis,UniversitdeGrenoble,1981. [50]N.Sbihi.Algorithmederecherched'unstabledecardinalitmaximum dansungraphesanstoile. DiscreteMathematics,29:5376,1980. [51]A.Schrijver. Combinatorialoptimization .SpringerVerlag,Berlin,2003. [52]F.B.Shepherd.Applyinglehman'stheoremstopackingproblems. MathematicalProgramming ,71(353367),1995. [53]G.Stauffer. Onthestablesetpolytopeofclaw-freegraphs.PhDthesis, EPFLausanne,2005. [54]G.Stauffer.Onthefacetsofthestablesetpolytopeofquasi-line graphs. OperationsResearchLetters,39(3):208212,2011. DiscussionColumn M anfredW.Padberg NodePackingsinGraphsandClawFree Graphs WhenVolkerKaibelcalledmeacoupleofweeksagotoaskmeto w riteashorthistoricalnote(asadiscussioncolumntothearticleby GianpaoloOriolo,GautierStauffer,andPaoloVentura)aboutnode packingsingraphsandespeciallyinclaw-freegraphs,myreaction wasmoreorless:Boy,ithasbeensomethinglikefortyyearsago thatI'veworkedonthatstuff!ButIpromisedtodoitandsohere itis.MostofwhatIhavetosayisaboutnodepackingsingraphs andthegeneralcontextinwhichithappened,butsomeofitmay helptoexplainhowtheinterestinclaw-freegraphstravelledfrom PittsburghviaBerlin,BonnandNewYorktoRome. 1Beginnings Hadyouaskedmebackin1969/1970aboutnodepackingorvertex p ackingingraphs(bull-free,claw-freeorwhatever),youwouldhave drawnablankstare:Iwouldnothaveknownwhatyouweretalking about.Thesetermsmayhaveexistedingraphtheory,butnotinintegerprogramming.Inintegerprogrammingwewereconcernedwith linearprogramswithpracticalapplications:airlinecrewscheduling problems,knapsackproblems(withtheirrootsincapitalrationing innance)andtravelingsalesmanproblems(TSPswiththeirroots inK.Menger'sBotenproblemfromthe1930's),tonamejustthree favoriteproblemsofmineinthosedays.Theintimateconnections betweengraphtheoreticalandintegerorzero-oneprogramming problemsthatyouyoungguysareallfamiliarwithtodayjust hadnotbeenestablishedyet.Allright,tounderstandTSPsyouneed amodicumofgraphtheory,butthat'sall.Whatgotmyinterestin thiseldwereairlinecrewschedulingproblems,perhapsbecause inSeptember1968IhadleftmynativeWestphaliainGermanyto ytoNewYorkandthen,bymyex-wife'scar,toPittsburgh,PA, whereIhadobtainedaFordFoundationFellowshiptocompletemy doctoralstudiesatCarnegie-MellonUniversity'sGSIAandwhereI specializedwithEgonBalas,seemyhistoricalnote Mixed-integerProgramming1968andthereafterin AnnalsofOperationsResearch 2007,149:163175 .DespiteDantzig,FulkersonandS.Johnson's milestone 1954paper SolutionofaLargeScaleTravellingSalesmanProblem publishedin OperationsResearch aworldrecordasitsolveda problemin1,128zero-onevariablestooptimalitycomputational integerprogramminghadabadname.EgonBalas'1965paper AnAdditiveAlgorithmforSolvingLinearProgrammingProgramswithZero-One Variables publishedin OperationsResearch hadbetteredthepicture somewhat,butallofthiswasovershadowedbyRalphGomory's algebraic,somesaidelegant,algorithmforintegerprogramming (rstabstractedin1958andpublishedin1963),whichcomputationallyjustdidnotwork.Ofthemanyreferencesintheliterature tothiseffect,letmejustmentionDonKnuth's1961paper MinimizingDrumLatencyTime inthe JournaloftheAssociationforComputing

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July2011 9 Machinery andsomearticlesinthebookeditedbyMuthandThompson IndustrialScheduling,Prentice-Hall,1963 .Inanycase,therecent revivalofGomory'smixed-integercutsincomputationalinteger programmingdoesnotcontradictwhatI'msayingbecausetheyare basedondisjunctivereasoningandthusdifferentfromtheoriginal Gomorycutsdeveloppedin1958orso;seealsomypaper Classical CutsforMixed-IntegerProgrammingandBranch-and-Cut in Math.Meth. Oper.Res.2001,53:173203 oritsreprintin AnnalsofO.R.2005, 139:321352 .SomuchforthehistoryasIfounditbackin1968in Pittsburgh,PA. 2StepI ByApril/Mayof1971IhadnishedanddefendedmyPhDthesis E ssaysinIntegerProgramming atGSIAandpreparedmyreturnto Germany,becauseofmyobligationtodosoundertheconditions ofmyFordFoundationFellowship.IwoundupattheInternational InstituteofManagementinBerlinwithathree-yearcontract.On March25,1970,EgonandIhadsubmittedChapter2ofmythesis OntheSet-CoveringProblem to OperationsResearch whereitappeared intheNovember-December1972issue.Chapter3ofmythesis 'Simple'Zero-OneProblems:SetCovering,MatchingsandCoveringsinGraphs waswidelydistributedasthe ManagementSciencesReportNo.235 of CMU'sGSIAandsubmittedto MathematicalProgramming sometime inlate1971.InitIhadabstractedfromEgon'sandmineresultsof Chapter2apropertyofthesaidproblemscalled`Simplicity'ofa polytope(whichImyselffoundinearly1972tobeerroneous). Inanycase,myChapter3containedtherstresultsonthefacets ofthesepolytopes,namelythecliqueand`lifted'odd-cyclefacets. Thepertainingcorrectresultsofitwerepublishedinsomewhatimprovedformin OntheFacialStructureofSetPackingPolyhedra,MathematicalProgramming,1973,5:199215 .Justforcompleteness,Chapter4ofmythesis EquivalentKnapsack-typeFormulationsofBounded IntegerLinearPrograms:AnAlternativeApproach appearedin NavalResearchLogisticsQuarterly,1972,19:699708 andoneofthetechnical appendices ARemarkonAnInequalityfortheNumberofLatticePoints inaSimplex in SIAMJournalofAppliedMath.,1971,20:638641 Anothertechnicalappendixofmythesiscontainedsomeresultson adjacentverticescuts,butthesewerejustagaincutsandnot facet-deningcutting-planesandthusIneverpublishedthatstuff. Voila,thatwasessentiallythecontentofmy1971thesis.Somuch forthosewhostillrecentlyaskedthemselveswhatmythesiswasall about.Justreadthepublishedstuff. 3StepII Ihadwantedtotestfacet-deningcutting-planesfornodean dset packingproblemscomputationallyinBerlin,buttherewerejustno adequatecomputingfacilitiesinBerlinandalsoalackoftestproblems.KarlaHoffmanandIdidsoeventuallyinourpaper SolvingAirline CrewSchedulingProblemsbyBranch-and-Cut publishedin Management Science,1993,39:657682 .Needlesstosay,itworkedwonderfully, buthereIamjumpingwayaheadoftime. IlandedajobatNewYorkUniversity'sGraduateSchoolofBusinessasofSeptember1974whenmyobligationtotheFordFoundationtostayinGermanywasover.OnmywayfromBerlinto NewYorkIstoppedforabouthalfayearatBernhardKorte's thennewInstituteforOperationsResearchandEconometricsat BonnUniversity,whereImetMartinGroetschel.Idon'tthinkthat itisnecessarytorecallourjointworkonthetravelingsalesman polytopehere.Besidesthetheoreticalwork,Martin's1976thesis containedthesolutiontooptimalityofa120-cityTSPusingonly facet-deningcutting-planesthat'salinearprogramin7,140zeroonevariablesandthusanotherworldrecord! Wunderbar ,because thatisexactlywhatIhadinmindwhenIstartedoutin1970to lookforfacet-deningcutting-planes,ratherthanarbitrarycuts withnoprovenmathematicalpropertiesotherthanvalidityfor theprobleminquestion.Iwon'trecalleitherindetailmycomputationalworkontheTSP(withS.Hong,thenwithH.Crowder andlaterwithGiovanniRinaldi)aswellasonotherproblemspursuedwiththesamegoaltoproveempiricallythevalueoffacetdeningcutting-planesinactualcomputation.ButIwillrecallmy workwithM.RamRao OddMinimumCut-Setsandb-Matchings,MathematicsofOperationsResearch,1982,7:67-80 ,whichwepresented atthe1979MathematicalProgrammingSymposiuminMontreal.For inthemeantime,thelateLeonidKhachianhadprovedthatlinear programmingproblemscouldbesolvedinpolynomialtimebythe ellipsoidmethodandhisresulthadtraveledtotheWestjustaround 1979.Aftermypresentation,JackEdmondsandLasloLovaszconjecturedthatRamandIhadjustgivenanotherpoly-timealgorithm forb-matchingsingraphs.ThisturnedouttobetrueandRamand IweredelightedwhenMartinGroetschelandOlafHollandshowed in Acutting-planealgorithmforminimumperfect2-matching,Computing, 1987,39:327344, thatourpurecutting-planealgorithm(usingthe simplexmethod,ofcourse,insteadoftheellipsoidalgorithm)outperformedEdmonds'graphicalalgorithminpracticalcomputation.I shouldaddthatpurelygraphicalproblemsoccuronlyrarelyinpracticeandarefrequentlycomplicatedbyadditionalconstraintssuchas capacityand/orcapitalconstraintswhichnecessitatesacutting-plane approach. ThemajorconsequenceofKhachian'sellipsoidmethodwasthe equivalenceof optimization and separation intermsofpoly-timesolvability,aresultthatwasobtainedinearly1980independentlyby threedifferentgroupsofresearchers:Groetschel,LovaszandSchrijver,KarpandPapadimitriouandPadbergandRao.(You'llnda proofofthisequivalence,e.g.,inChapter9ofmybook LinearOptimizationandExtensions,1995,2nded.1999,SpringerVerlag ).This equivalencegeneralized,ofcourse,EdmondsandLovasz'sconjecturementionedabove.Butitalsoreinforcedthehuntforfacetdeningcutting-planesthathadfollowedmyinitialndingsthat facets oftheconvexhulloftheintegersolutionscouldindeedbefound for(some)integerandmixed-integerprograms,likenodecovering, nodepacking,setpackingproblemsandthenknapsackproblems, TSPs,etc. 4StepIII Alsoinoraround1979IlearnedthatthelateGeorgeMintyhad g eneralizedEdmonds'matchingalgorithmandfoundapoly-timealgorithmforvertexpackingsinclaw-freegraphs,see JournalofCombinatorialTheoryB,1980,284-304 ,andindependentlyofhimNajiba Sbihi,see AlgorithmedeRecherched'unStabledeCardinaliteMaximum dansunGraphesansEtoile,DiscreteMathematics,1980,29:5376 aswell.Oncetheequivalenceofoptimizationandseparationhad beenestablished,giventhepoly-timesolvabilityofweightedvertex packingin K 1 ;3 -freegraphs,theseparationproblemfortheassociatedconvexhullofnodepackingsinsuchgraphshadtobesolvable inpoly-timeaswell.Beinganeternaloptimist,Iputtheproblemof ndingacompleteminimallineardescriptionforthisproblemonmy listofthingstodo,butnevercamearoundtoattackingthisproblem alone. Sometimein1981/1982,thelateMarioLucertiniofRome's UniversitaTorVergata invitedmetodoatwo-weekintensivecourseon combinatorialoptimizationinRomeitmusthavebeeninAugust of1982,becauseitwasawfullyhotandtheclassroomhadnoairconditionning.ImetthroughLucertiniGiovanniRinaldiandAntonio Sassano,whohadjustre-joinedtheItalianCNRafterhavingworked

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10 OPTIMA86 forawhileintheirowncompany.Afteroneofmycourses,thefo ur ofusdiscussedoveracoolbeerinoneoftheshadysquaresof RomeapossiblevisitofAntonioandGiovanniwithmeatNewYork University.AntonioSassanocamein1983/1984toworkwithmeat NYU,Giovannicamealittlewhilelater.IsuggestedtoAntonioto workonthefacialstructureofthepolytopeofvertexpackingsin K 1 ;3 -freegraphs,butperhapsduetotherelativelyshorttimethat AntoniostayedwithmeinNewYork,wedidnotgetenoughsubstantialresultsontheproblemtowriteajointpaperonit.When hereturnedtoRome,hetooktheproblemalongwithhimandthe desiretosolveit;justlookatSassano'shomepageatthe Universita LaSapienza,whereseveralpapersonthistopic(withvariouscoauthors)arelisted.IamsurethatAntonioSassanohasalottodowith theprogressmadeinItalyonthisproblemlikethenewalgorithmby Faenza,Oriolo,andStaufferandthenewpolyhedralresultsbyGalluccio,Gentile,andVentura.Personnally,whatIndveryinteresting is,ofcourse,apurecutting-planealgorithmforthisproblem,like theonethatRamRaoandIfoundforb-matchingsingraphsandthat ( simplexmethodorellipsoidmethod,Idon'tcare)iscomputationally efcientandpermitsothercomplicatingconstraintstobeadded.I amhappytohearthattheseparationroutineviathenewextended formulationduetoFaenza,Oriolo,andStaufferprovidesthis. ManfredW.Padberg,EmeritusProfessor,NewYorkUniversity,SternSchool ofBusiness,17rueVendme13007Marseille,France manfred4@wanadoo.fr Obituary O legBurdakov,JohnDennis,andJorgeMor CharlesG.Broyden,1933011 CharlesGeorgeBroyden, S weden,2002(Photo: OlegBurdakov) CharlesGeorgeBroydenwasborn February2nd,1933,inEngland.He receivedhisdegreeinPhysicsfrom KingsCollegeLondonin1955.He spentthersttenyearsofhiscareerinindustry.In1967,hemoved totheUniversityofEssexwhere hebecameaprofessorand,later, deanoftheSchoolofMathematics.In1986,hedecidedtoretire earlytobecomeatravelingscholar, butin1990,heacceptedanappointmentasaprofessorofnumericalanalysisattheUniversityof Bologna. Charlesreceivedinternational recognitionforhisseminal1965 paperinMathematicsofComputation,inwhichheproposedtwomethodsforsolvingsystemsofequations.TheylaterbecameknownasBroyden'smethods.Anotherof hismostimportantachievementswasthederivationoftheBroydenFletcher-Goldfarb-Shanno(BFGS)updatingformula,oneofthekey toolsusedinoptimization.Moreover,Charleswasamongthosewho derivedthesymmetricrank-oneupdatingformula,andhisnameis alsoattributedtotheBroydenfamilyofquasi-Newtonmethods. AtBologna,Charlesshiftedthefocusofhisresearchtonumerical linearalgebraand,inparticular,toconjugategradientmethodsand tothetaxonomyofthesemethods.Someofthemainresultsofthat periodaresummarizedinhis2004bookwithM.T.Vespucci Krylov SolversforLinearAlgebraicSystems Inrecognitionofhisfundamentalcontributionstothedevelopmentofoptimizationandnumericalmathematics,thejournal OptimizationMethodsandSoftware (OMS)establishedtheCharlesBroydenprize.Itisawardedyearlyforthebestpaperpublishedin OMS. WhenCharleslearnedabouttheprize,hemodestlynotedthat IdiscoveredmyalgorithmsbecauseIwasintherightplaceatthe righttime.Beingintherightplaceattherighttimeoncecouldbe goodluck,butifthishappensseveraltimes,thisclearlyindicatestalent.Indeed,onecanhardlyndabookonnumericaloptimization wherethediscoveriesofCharlesBroydenarenotmentioned. CharlesBroydendiedonMay20,2011.Wewillrememberhimas ahighlydedicated,modest,andhonestresearcher,respectedbyhis manyfriendsandcollaboratorsaroundtheworld.Weexpressour sympathytohiswifeJoan,hischildrenandgrandchildren. Announcements C allforNominationsforthe2012 Beale-Orchard-HaysPrize Nominationsareinvitedforthe2012Beale-Orchard-HaysPri zefor excellenceincomputationalmathematicalprogrammingthatwillbe awardedattheInternationalSymposiumonMathematicalProgrammingtobeheldinBerlininAugust2012. ThePrizeissponsoredbytheMathematicalOptimizationSociety, inmemoryofMartinBealeandWilliamOrchard-Hays,pioneersin computationalmathematicalprogramming.Nominatedworksmust havebeenpublishedbetweenJan1,2009andDec31,2011,and demonstrateexcellenceinanyaspectofcomputationalmathematicalprogramming.Computationalmathematicalprogrammingincludesthedevelopmentofhigh-qualitymathematicalprogramming algorithmsandsoftware,theexperimentalevaluationofmathematicalprogrammingalgorithms,andthedevelopmentofnewmethods fortheempiricaltestingofmathematicalprogrammingtechniques. Fulldetailsofprizerulesandeligibilityrequirementscanbefoundat http://www.mathopt.org/?nav=boh T he2012PrizeCommitteeconsistsofMichaelFerris(Chair), PhilipGill,TimKelley,andJonLee. Nominationscanbesubmittedelectronicallyorinwriting,and shouldincludedetailedpublicationdetailsofthenominatedwork. Electronicsubmissionsshouldincludeanattachmentwiththenal publishedversionofthenominatedwork.Ifdoneinwriting,submissionsshouldincludefourcopiesofthenominatedwork.Supporting justicationandanysupplementarymaterialarestronglyencouraged butnotmandatory.ThePrizeCommitteereservestherighttorequestfurthersupportingmaterialandjusticationfromthenominees. Nominationsshouldbesubmittedto: Prof.MichaelFerris,ComputerSciencesDepartment,University ofWisconsin,1210WestDaytonStreet,Madison,WI53706,USA Email: ferris@cs.wisc.edu ThedeadlineforreceiptofnominationsisJanuary15,2012.

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July2011 1 1 ISMP2012inBerlin The 21stInternationalSymposiumonMathematicalProgramming(ISMP 2012) willtakeplaceinBerlin,Germany,August1924,2012.ISMP isascienticmeetingheldeverythreeyearsonbehalfoftheMathematicalOptimizationSociety(MOS).Itistheworldcongressof mathematicaloptimizationwherescientistsaswellasindustrialusers ofmathematicaloptimizationmeetinordertopresentthemostrecentdevelopmentsandresultsandtodiscussnewchallengesfrom theoryandpractice. ConferenceTopics Theconferencetopicsaddressalltheoretical,computation aland practicalaspectsofmathematicaloptimizationincluding: integer,linear,nonlinear,semidenite,conicandconstrainedprogramming discreteandcombinatorialoptimization matroids,graphs,gametheory,networkoptimization nonsmooth,convex,robust,stochastic,PDE-constrainedand globaloptimization variationalanalysis,complementarityandvariationalinequalities sparse,derivative-freeandsimulation-basedoptimization implementationsandsoftware operationsresearch logistics,trafcandtransportation,telecommunications,energy systems,nanceandeconomics ConferenceVenue TheSymposiumwilltakeplaceatthemainbuildingofTUBerlin in theheartofthecityclosetotheTiergartenpark. TheopeningceremonywilltakeplaceonSunday,August19,2012, attheKonzerthausonthehistoricGendarmenmarktwhichisconsideredoneofthemostbeautifulsquaresinEurope.Theopening sessionwillfeaturethepresentationofawardsbytheMathematicalOptimizationSocietyaccompaniedbysymphonicmusic.Thisis followedbythewelcomereceptionwithamagnicentviewonGendarmenmarkt. TheconferencedinnerwilltakeplaceattheHausderKulturen derWelt(HouseoftheCulturesoftheWorld)locatedinthe TiergartenparkwithabeergardenontheshoresoftheSpreeriver andaviewontheGermanChancellery. RegistrationandImportantDates TheconferenceregistrationwillopenbeforeDecember2011. The abstractsubmissiondeadlinewillbeApril15,2012,theearlyregistrationdeadlineJune15,2012. InaccordancewiththenewMOSmembershipfeespolicy,ISMP 2012willofferthreeearlyregistrationratesforregularattendees (notstudents,notretired,andnotlifetimemembersofMOS): Euro340includingMOSmembershipfor2013 Euro390includingMOSmembershipfor2013and2014 Euro415includingMOSmembershipfor20132015 Theearlyregistrationratesforretirees(notlifetimemembersof MOS)are Euro190includingMOSmembershipfor2013 Euro215includingMOSmembershipfor2013and2014 Euro230includingMOSmembershipfor20132015 TheearlyregistrationrateforstudentsisEuro160.TheearlyregistrationrateforlifetimemembersofMOSisEuro280.Theregistrationratesforlateregistration(afterJune15,2012)willbehigher (detailstobeannounced). Webpageetc. Moredetails(includingregistrationrates,hotelprices,s ponsorship opportunities,exhibitsetc.)canbefoundontheconferenceweb pagesat www.ismp2012.org MainbuildingofTUBerlinintheheartofthecityclosetotheT iergartenpark (Photo:TUBerlin/Dahl) HausderKulturenderWelt(HouseoftheCulturesoftheWorld )locatedinthe Tiergartenpark(Photo:ChristophEyrich)

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12 OPTIMA86 C allforSitePre-Proposals ISMP2015 TheSymposiumAdvisoryCommitteeoftheMathematicalOptim izationSocietyissuesacallforpre-proposalstoorganizeandhost ISMP2015,thetriennialInternationalSymposiumonMathematical Programming. ISMPistheagshipeventofoursociety,regularlygatheringover athousandscientistsfromaroundtheworld.Theconferencewillbe heldinoraroundthemonthofAugust,2015.HostingISMPprovides avitalservicetothemathematicaloptimizationcommunityandoftenhasalastingeffectonthevisibilityofthehostinginstitution.It alsopresentsasignicantchallenge.Thiscallforpre-proposalsis addressedatlocalgroupswillingtotakeupthatchallenge. PreliminarybidswillbeexaminedbytheSymposiumAdvisory Committee(SAC),whichwillthenissueinvitationsfordetailedbids. ThenaldecisionwillbemadeandannouncedduringISMP2012in Berlin.MemberoftheSACare JeffLinderoth,Chair linderoth@wisc.edu S habbirAhmed, sahmed@isye.gatech.edu K urtAnstreicher, kurt-anstreicher@uiowa.edu A ndersForsgren, andersf@kth.se M artineLabbe, mlabbe@ulb.ac.be R digerSchultz ruediger.schultz@uni-due.de ,and S huzhongZhang, zhangs@umn.edu P reliminarybidsshouldbebriefandcontaininformationpertainingtothe (1)Location, (2)Facilities, (3)Logistics:Accommodationandtransportation,and (4)Likelylocalorganizers. Furtherinformationcanbeobtainedfromanymemberofthe advisorycommittee.Pleaseaddressyourpreliminarybidsuntil September15,2011toJeffLinderoth linderoth@wisc.edu ConicOptimizationSpecialIssuefor P acicJournalofOPTIMIZATION Originalresearcharticlesaresolicitedforaforthcomingi ssueof PacicJournalofOPTIMIZATION dedicatedtoConicOptimization. Potentialarticlesmayfocuson(orrelateto)anysubsetofthe theoretical,computationalandpracticalaspectsofConicOptimization. Overthelasttwodecades,therehasbeenasignicantadvance intheresearchofinteriorpointmethodsandconicoptimization. Todayconicoptimizationhasemergedasamajorcomputational paradigmandhasmadesignicantimpactonnumerousproblems previouslyconsideredintractableordifculttoapproximate.This specialissueaimstosolicitpapersthatprovideoriginalresearchon conicoptimizationtheoryandalgorithms,includingavailablesoftware.Alsowelcomearepapersthatdealwithemerging,meaningfulapplications;orthatgivefriendlyoverviewsofcertaintheoreticallyadvancedconicoptimizationtechniquesrelevanttoapplications. Examplesoftopicsthatwillbeaddressedinthisspecialissueinclude,butarenotlimitedto: Theoryandalgorithmsforlargescaleconicoptimization Computationalstudyofconicoptimizationalgorithms Relaxationtechniquesbasedonconicoptimization Applicationsofconicoptimizationinscienceandengineering (e.g.,imageprocessing,compressedsensing,digitalcommunicationsandnetworks) Instructionsforauthorsareavailableontheweb: http://www.ybook.co.jp/online/forauthors.htm Pleasee-mailthepdf-leofyourfullmanuscriptwithacover messageinplaintexttooneoftheguesteditorsby September1,2011 Zhi-QuanLuo,UniversityofMinnesota, luozq@umn.edu LeventTunel,UniversityofWaterloo, ltuncel@math.uwaterloo.ca NaihuaXiu,BeijingJiaotongUniversity, nhxiu@center.njtu.edu.cn IMPRINT E ditor: KatyaScheinberg,DepartmentofIndustrialandSystemsEngineering,LehighUniversity,Bethlehem,PA,USA. katyascheinberg@gmail.com Co-Editors: S amuelBurer,DepartmentofManagementSciences,TheUniversityofIowa,IowaCity,IA52242-1994,USA. samuel-burer@uiowa.edu VolkerKaibel,Institut f rMathematischeOptimierung,Otto-von-GuerickeUniversittMagdeburg,Universittsplatz2,39108Magdeburg,Germany. kaibel@ovgu.de FoundingEditor: D onaldW.Hearn PublishedbytheMathematicalOptimizationSociety. DesignandtypesettingbyChristophEyrich,Berlin,Germany optima@0x45.de PrintedbyOktoberdruckAG,Berlin,Germany. ApplicationforMembership I wishtoenrollasamemberoftheSociety.Mysubscriptionisformypersonaluse andnotforthebenetofanylibraryorinstitution. Iwillpaymymembershipduesonreceiptofyourinvoice. Iwishtopaybycreditcard(Master/EuroorVisa). Creditcardno. Expirationdate Familyname Mailingaddress Telephoneno. Telefaxno. E-mail Signature M ailto: MathematicalOptimizationSociety 3600MarketSt,6thFloor Philadelphia,PA19104-2688 USA Chequesormoneyordersshouldbemade payabletoTheMathematicalOptimization Society,Inc.Duesfor2011,includingsubscriptiontothejournal MathematicalProgramming ,areUS$90.Retiredare$45. Studentapplications:Duesare$22.50. Haveafacultymemberverifyyourstudent statusandsendapplicationwithduesto aboveaddress. Facultyverifyingstatus Institution


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OPTIMA

Mathematical Optimization Society Newsletter




Philippe L. Toint


MOS Chair's Column
July I, 2011 I. For the few sorry spirits who, misguided by the gen-
eral state of the world, thought that mathematical optimization was
in theoretical decline or too remote from applications in the real
world, the SIOPT Conference on Optimization in Darmstadt in May
was a real and vivid counterexample. Indeed this very well attended
meeting (600+ participants) was a resounding success in terms of
interest and quality of the talks. As has been so far the case in this
series of meetings, the focus was mainly on continuous problems:
in particular, problems arising from continuous mechanics, fluids and
control were prominent, showing the very healthy state of not only
optimization in those domains, but also the German industry's inter-
est in optimization in general. The significant presence of discrete
optimization was also noticeable, with several interesting sessions
and plenary talks in this area. As optimizers, all were delighted that,
once more, a high quality conference has been organized by SIAM
in addition to the major events organized by the Mathematical Op-
timization Society.
If mathematical optimization is turning today into one of the major
branches in applied mathematics, this is due not only to our present
efforts as scientists, but also to those of the founding fathers of our
research domain. One of them, Charles Broyden (the B in BFGS)
unfortunately passed away on Friday 20th May, at the age of 78 (see
the obituary published in this issue on page 10). His memory will
stay with us for long, and his work will undoubtedly continue to
inspire.
The beginning of 201 I was also the time to start thinking about
the various prizes sponsored by MOS, which will be awarded in the
International Mathematical Programming Symposium in Berlin in Au-
gust 2012. It may be useful to recall that the MOS currently awards
five scientific prizes and a named lectureship. These are the Dantzig,
Lagrange, Beal-Orchard-Hays, Fulkerson and Tucker prizes, and the
Paul Tseng Lectureship, whose more complete description, scope
and past winners can be found on the MOS Website (http://www.
mathprog.org). The respective committees have now been estab-
lished for deciding to whom these distinctions must be given, and
I would like to take this opportunity to thank all of our colleagues
who kindly accepted to serve on these committees. I would also like
to call on all members to think about proposing high quality submis-
sions for these prizes. I am certain that their scientific value can only
be enhanced by friendly competition between high quality submis-
sions. I am personally looking forward to meeting you all at the the
award ceremony during the opening session of the Berlin ISMP.
This is also the time to start looking at possible sites that will host
ISMP in 2015. The call for proposal submission can be found in this
issue on page 12. And as always, do not forget to renew your MOS
membership.
Meanwhile, enjoy the summer (for the majority of us in the north-
ern hemisphere) and let us keep the abundance and quality of our
scientific activities at the present vibrant level.


Note from the Editors
The stable set problem in claw-free graphs is the main topic of this
issue of Optima. Much of the tremendous progress that has recently
been obtained on this generalization of the matching problem is
due to work of the authors Gianpaolo Oriolo, Gautier Stauffer, and
Paolo Ventura of the article you'll find below and their co-workers.
In the discussion column, Manfred Padberg shares with us his mem-
ories of the historical context in which the interest in the stable set
problem in claw-free graphs arose and of how it traveled to Italy.
Katya Scheinberg, Editor
Sam Burer, Co-Editor
Volker Kaibel, Co-Editor



Contents of Issue 86 / July 201 I
I Philippe L. Toint, MOS Chair's Column
I Note from the Editors
I Gianpaolo Oriolo, Gautier Stauffer and Paolo Ventura,
Stable Sets in Claw-Free Graphs: Recent Achievements and
Future Challenges
8 Manfred W. Padberg, Node Packings in Graphs and Claw Free
Graphs
10 Oleg Burdakov, John Dennis, and Jorge More,
Charles G. Broyden, 1933-201 I
10 Call for Nominations for the 2012 Beale-Orchard-Hays Prize
II ISMP 2012 in Berlin
12 Imprint





Gianpaolo Oriolo, Gautier Stauffer and Paolo Ventura
Stable Sets in Claw-Free Graphs: Recent
Achievements and Future Challenges



I Introduction
A stable set in a graph G(V,E) is a set of vertices that are pairwise
non-adjacent. When G is the intersection graph of the edges of a
graph H two edges intersect if they share an endpoint a stable
set in G corresponds to a matching in H (and vice-versa). Hence
while the stable set problem is hard in general, the special case of
line graphs the family of all such intersection graphs can be han-
dled in polynomial time through matching.
Matching is a classic problem in combinatorial optimization and it
exhibits some remarkable properties. Many of those properties have






OPTIMA 86


been extended and led to very powerful tools and theories like for
instance matroid intersection or delta-matroids. In order to extend
the matching theory to the stable set setting, it appears that two
fundamental properties of matching are crucial: the augmenting path
property and the intersection property.
Petersen observed in 1891 (and Berge proved in 1952) that the
symmetric difference of two matching is made of alternating paths
and even cycles. In particular, a matching M is of maximum cardinality
in a graph G if and only if there does not exist any augmenting path
in G with respect to M. Moreover, as two matching are adjacent on
the matching polytope MATCH(G) the convex hull of all incidence
vectors of matching in a graph G if and only if they have a con-
nected symmetric difference, one can easily show that MATCH(G)
has the intersection property: MATCH(G) n {x : eeE EXe = k} is
integral for every integer k.
Interestingly those properties extend to the stable set setting be-
yond line graphs (alternating paths and cycles being defined in terms
of vertices here): they are also valid for stable sets in claw-free graphs
- a graph is claw-free if no vertex has a stable set of size three in
its neighborhood. This was observed by Berge in 1973 for the sym-
metric difference of stable sets in claw-free graphs and by Calvillo in
1979 for the intersection property. Remarkably, Berge and Calvillo
also proved the converse, i.e., a class of graphs exhibits one or the
other of those properties for the stable set problem if and only if
it is a subclass of claw-free graphs. Hence, with respect to stable
sets, claw-free graphs appear to be the right framework to extend
the aforementioned properties of matching. The problem of finding
a maximum weighted stable set in claw-free graphs has been there-
fore investigated by several people, and its theory has been develop-
ing for more than 40 years. The last 10 years have been particularly
productive, mainly due to new approaches that exploit results from
structural graph theory. The purpose of this paper is to help the in-
terested researchers to navigate through the various results in this
field and in particular to shed light on the latest achievements and
the current open questions.
For the sake of shortness, some theorems might be slightly imprecise.
In this case, a reference is given, and the reader should rely on that. Also
we often denote by V(G) and E(G) the vertex set and the edge set of a
graph G.


2 Stable Sets in Claw-Free Graphs: Some Classical
Results
In this section we survey a few classical results on the problem. We
first deal with some algorithmic results, and then move to some
polyhedral questions.

2. 1 Algorithms for the Maximum Weighted Stable Set Problem
Given a claw-free graph G(V,E) and a weight function w : V R,
a maximum weighted stable set (MWSS) can be found in polynomial
time. We denote by a(G) the cardinality of such a stable set when
w is the all ones vector; a(G) is also called the stability number of G.
At the present time, there are several algorithms for the problem,
and we may recognize three different main approaches. A first class
of algorithms deals with augmenting paths techniques, and the algo-
rithms by Minty [35] and Sbihi [50], respectively, for the weighted
and the unweighted case, follow this approach. In fact, as we already
discussed, Berge's augmenting path theorem for matching extends
to stable sets in claw-free graphs (a path P is augmenting with re-
spect to a stable set S if (V(P) \ S) u (S \ V(P)) is a stable set of
size ISI+1):
Theorem I ([4]). A stable set S is maximum for a claw-free graph G
if and only if there are no paths that are augmenting with respect to S.


Sbihi's algorithm builds upon this theorem while Minty's builds
upon a cute extension to the weighted case (given an augmenting
path P with respect to a stable set S, the weight of this path is given
by w(V(P) \S) w(V(P) n S)):

Theorem 2 ([35]). Let S be a MWSS of size k, and let P be an aug-
menting path of maximum weight with respect to S. Then (S \ V(P)) u
(V(P) \ S) is a MWSS of size k + 1.

Minty's idea is to detect those maximum weight augmenting paths
and proceed with at most I VI augmentations. Given two "exposed"
vertices u, v of V \ S, i.e., they are both adjacent to a single vertex
of S, Minty's crucial idea is that of reducing the problem of finding
an u v augmenting path with maximum weight to the problem of
finding a matching with maximum weight in an auxiliary graph H. The
construction of H is rather intricate. We simply mention here that
this graph has (I|V|I) vertices. Hence the whole algorithm requires
the solution of O(|V|3) weighted matching problems in an auxiliary
graph with O(IVI) vertices.
In 2001 the algorithm of Minty was slightly revised by Naka-
mura and Tamura [36], as they realized that, in the weighted case,
the algorithm could fail for some special configurations. Subse-
quently, Schrijver [51], elaborating on Minty's algorithm, proposed
an elegant alternative using a slightly different edge-weighted aux-
iliary graph H. The algorithm can be implemented to run in time
O(IVIs log IVI+IVI4IEI) in the weighted case and in time O(|IV|) in
the unweighted one (however, Sbihi claimed that her algorithm, for
the unweighted case, can be implemented to run in time O(IV13).
An entirely different approach, based on reduction techniques,
was taken by Lovasz and Plummer [34], for solving the problem in
the unweighted case. The crucial idea here is that of performing a
series of graph reductions that preserve the stability number, as to
end up with a line graph, where one has to solve a single matching
problem. The resulting algorithm is very elegant, much less intricate
than the previous algorithms, and, as Lovasz and Plummer point out,
with some care it can be implemented as to run in O(|V|14). Unfor-
tunately, in spite of some efforts, it is not clear how to extend this
algorithm to the weighted case.
However, recently Nobili and Sassano [39] were able to combine
ideas from both the algorithm of Minty and that of Lovasz and Plum-
mer to provide a new algorithm for the weighted case that runs in
O ( V log I V)-time. If we compare (very roughly!) their algorithm
with Minty's algorithm, we see that, on one hand Nobili and Sas-
sano are able to reduce the number of matching problems that have
to be solved to O(IV12), while on the other they are able to solve
each of these problem in O ( V12 log I VI )-time, thanks to a weighted
reduction, inspired from that of Lovasz and Plummer.
A latter solution approach to the MWSS problem in claw-free
graphs is based on decomposition techniques and has been taken
by Oriolo, Pietropaoli and Stauffer [41] first, and by Faenza, Oriolo,
and Stauffer [19] later. The latter algorithm can be implemented to
run in time O(IVI (IVI log|V + IEI)). We postpone the discussion
about these algorithms to Section 4.1, as it is first convenient to deal
with some structural decomposition results for claw-free graphs.

2.2 Stable Sets in Claw-Free Graphs: Polyhedral Issues
The stable set polytope STAB(G) of a graph G(V,E) is the convex
hull of the characteristic vectors of stable sets in G, i.e., STAB(G)
conv{x e {0, 1}1 I xu+Xv, < 1, V {u, v} E}. Since the seminal
paper by Padberg [43], this polytope has been carefully investigated
by several authors (see e.g. [I 12, 37, 38]).
Because the MWSS problem in claw-free graphs can be solved in
polynomial time, exact separation over this polytope also can be
done in polynomial time [28], and hence the stable set polytope of






July 201 I


claw-free graphs is somewhat "under control". However no com-
plete linear description is known at the time of writing, despite the
fact that the problem is "officially" open for more than a quarter of
a century [29]: "in spite of considerable efforts, no decent system
of inequalities describing "STAB(G)" for claw-free graphs is known".
Such a description would possibly result in a nice minmax character-
ization of the problem.
A neat description is at hand for the stable set polytope
of line graphs. Indeed, Edmonds [15] proved that the match-
ing polytope MATCH(H) of a graph H(V,E) i.e., conv{x e
{0, 1} El e 6(v) xe < 1, Vv e V} can be described by non-
negativity inequalities, degree inequalities (as usual, we denote by
6(v) the set of edges incident to a node v), and odd sets inequal-
ities, where, for an odd set S c V, we denote by E(S) the set of
edges between vertices of S.

Theorem 3 ([15]). The matching polytope of a graph H(V,E) can
be characterized as MATCH(H) = x e GRE | x > 0; lec6(v) Xe <
1, Vv e V; ZeeF(S) Xe < [ ]}, for every odd set S c V}.

But since MATCH(H) = STAB(L(H)), where L(H) denotes the
line graph of H, it follows that the stable set polytope of line graphs
can be described by non-negativity inequalities, clique inequalities
and Edmonds' inequalities, the counterpart of odd set inequalities in
the stable set setting. More formally:

Definition 2.1. For a graph G(V,E) and an odd set of cliques K, let
V>2 (K) be the set of vertices covered by at least 2 cliques of K. The
Edmonds' inequality associated with K is: YvCv>2(x) xv < [ 2J.

From Definition 2.1 it follows that Edmonds' inequalities are de-
rived as Chvatal-Gomory cuts from the clique relaxation of the stable
set polytope QSTAB(G) : {x e I vR : x > 0;x(K) < 1, for every
clique K of G} [17], i.e., they can be obtained by first taking a non-
negative combination of the inequalities describing QSTAB(G), and
then rounding down the right hand side of the combination.

Lemma 2.1 ([I 15]). For a line graph G, non-negativity inequalities,
clique inequalities and Edmonds' inequalities are enough to describe the
stable set polytope.

Unfortunately, Lemma 2.1 does not hold true for claw-free
graphs. In fact, consider a 5-wheel, i.e., a graph with vertex set
{w, V, V2, V3, V4, V5} and edge set {(w, vi), (vi, vi+l) for all i
1,.., 5} with v6 = vi, then the 5-wheel inequality x,, + 2xw < 2
is a facet of its stable set polytope. This shows that non-rank in-
equalities are needed in order to define the stable set polytope
of claw-free graphs. An inequality is rank if it only involves {0,1}-
valued coefficients in the left hand side, i.e., if it is of the form
vsXv < oc(G[S]) forS c V.
In 1978 Maurras, inferring that 5-wheels and, more generally,
odd-antiwheels (i.e., a graph made of a vertex totally joined to the
complement of an odd hole) were the problem, introduced the class
of quasi-line graphs, i.e., claw-free graphs without odd-antiwheels. He
also conjectured that for quasi-line graphs all facets of STAB(G) are
rank. Building upon Maurras' conjecture, Sbihi conjectured that for
claw-free graphs all facets of STAB(G) have only {0,1,2}-valued co-
efficients. Both conjectures were proven false by Giles and Trotter
[27] in 1981 (see Figure 2 and I, respectively). We know now that
for claw-free graphs with stability number 3 there exist facets with
arbitrarily many coefficients [46] and that for any integer a there ex-
ist quasi-line graphs whose stable set polytopes involve facets with
coefficient a and a + 1 [27, 33]. While Maurras' conjecture was
wrong, his intuition on the relevance of the class of quasi-line graphs
was correct. Indeed, in contrast with general claw-free graphs,


3 7
Figure I. The complement of a claw-free graph G. The graph G induces the
facet: 2xi + 2x2 + 2x3 + 2x4 + 2x5 + x6 + x7 + 3xg + 3xg + 3x10 < 4. Note
that G is not quasi-line and that oc(G) = 3. (The picture is a courtesy of Tristram
Bogart, Annie Raymond and Rekha Thomas.)



the nature of the inequalities needed for quasi-line graphs was
grasped first by Ben Rebea [49] and later by Oriolo [40] who named
a conjecture after him: the Ben Rebea conjecture.

The Ben Rebea Conjecture I (Oriolo [40]). For a quasi-line graph
G, non-negativity inequalities, clique inequalities and clique family inequal-
ities are enough to describe STAB(G).

Definition 2.2. Given a graph G, a family of cliques K and an integer
p > 2, define V>p(K) and Vp 1(K) as the set of vertices covered by
at least p cliques and exactly p 1 cliques, respectively. The following
inequality is valid for STAB(G) and is called the clique family inequality
associated with X and p: v v>,p(x) xv + P r Svevp_ (x) Xv <
[ J, where r X1 mod p.

Clique family inequalities generalize Edmonds' inequalities,
and their validity can easily be derived by the disjunction
YVCV,(x)Ur, x) x <, I P I V YaCVP(x)Ur e x, >I \X I +
1 applied to QSTAB(G).
The Ben Rebea conjecture suggested that the stable set polytope
of quasi-line graphs has a neat description. As for claw-free graphs,
in 1991 Galluccio and Sassano [26] provided an elegant characteri-
zation of rank minimal facets, i.e., rank facets that are minimal with
respect to lifting and complete join operations [I I, 43].
We close this section by illustrating the result of Calvillo [5] that
we mentioned before. Calvillo proved the following nice property of
the stable set polytope of claw-free graphs. A polytope P c Rn has
the intersection property if P n {x e R1 : 1 xi = k is integral for
all integer k.

Theorem 4 ([5]). STAB(G) has the intersection property if and only
if G is a claw-free graph.


Figure 2. A quasi-line graph inducing the facet ZvGo xv + 2 *- Y,. xv < 6. On
the right, the cliques involved in the derivation of the inequality as a clique family
inequality.






OPTIMA 86


3 A Breakthrough: Decomposition of Claw-Free and
Quasi-Line Graphs
In a long series of paper, Chudnovsky and Seymour (see e.g.
[8, 9, 10, 6]) elucidate the structure of claw-free graphs and de-
fine a decomposition result for them. For this purpose, they have
introduced a new composition operation.

3.1 The Composition of Strips
In order to better grasp this operation, it is convenient to first deal
with an algorithmic procedure that can be used to build line graphs.
The rationale of this latter operation is the following. Given a graph
G, each vertex in G can be associated with a clique in the line graph
H L(G) (all edges incident to this vertex are pairwise adjacent in
H). If we let F denote the family of cliques of H that are associated
with vertices of G, we observe that F has the following properties:
(i) every edge of H is covered by some clique of f; (ii) every vertex
of H is covered by exactly two cliques of F.
Suppose now that we are given a (general) graph H. We call a
family f of cliques of H a Krausz family if it satisfies the above prop-
erties. Krausz [32] proved the following:
Theorem 5 ([32]). A graph is the line graph of a multi-graph if and
only if it admits a Krausz family.
This theorem gives an algorithmic procedure to build line graphs.
This procedure requires as input a set of vertices V and a par-
tition P Pi,...,Pq of the multi-set V u V. It then associates
to the pair (V, P) the graph G with vertex set V and edge set
E : {{u, v} : u # v and both u,v e Pi, for some 1 < i < q}.
Chudnovsky and Seymour generalized the above construction, es-
sentially by replacing vertices with strips. We borrow (but slightly
change) some definitions of theirs.
Definition 3.1. A strip (G, A) is a graph G (not necessarily con-
nected) with a multi-family A of either one or two designated non-empty
cliques (possibly identical) of G. The cliques in A are called the extremi-
ties of the strip.
Let H- = {(GZ, A), i 1= ...,k} be a family of vertex disjoint
strips. Let A(3-) denote the multifamily of the extremities of those
strips, i.e., A(H) Ui1..k-Ai, and let T Pi,P2,..., Pq be the
classes of a partition of A (H). We associate to the pair (H, P) the
graph G that is made of the disjoint union of the graphs G1, ... Gk,
with additional edges E : {{u, v} : u v and u and v belong to
different extremities in a same class Pi, for some 1 < i < q}. G is
called the composition of the strips with respect to partition P.
Note that, for line graphs, this composition reduces to the above
construction, as soon as each graph Gi is made of a single vertex vi
and the corresponding strip is ({ vi, {{vi}, {vi}}).
Even though the operation of composition of strips builds graphs
that are in general non-line, such graphs indeed inherit a "line struc-
ture" from its similarity with Krausz composition. Say that a strip
H (G, A) is line if G admits a Krausz family K with A c K. Then,
as soon as all strips are line, the composition is a line graph. The
proof of this fact is straightforward. We will make heavy use of this
fact in the following.
Lemma 3.1. Let G be the composition of a family of line strips
Hi = (Gi, Ai), = 1,...,k with respect to a partition P. Then G is
a line graph.

3.2 Decomposition Results for Claw-Free and Quasi-Line Graphs
In [8] Chudnovsky and Seymour overview a series of papers in which
they prove a structure theory for claw-free graphs. The theory is too
complex to describe in detail here, so we just outline two of their
results.


Theorem 6 ([10]). Let G (V, E) be a connected claw-free graph. Then
one of the followings holds: i) ac(G) < 3 and G belongs to a small set of
basic graphs; ii) G is a fuzzy circular interval graph; iii) G is the compo-
sition of strips, that are either fuzzy linear interval strips or they belong
to one of a small number of family of strips, all with stability number at
most 3.

Circular interval graphs are defined by a set of vertices, a circle and
a set of arcs. Vertices are mapped to the circle and two vertices are
adjacent if and only if they are covered by an arc. Those graphs are
also known as proper circular arc graphs. Linear interval graphs are
constructed in the same way as circular interval graphs, but on a line
rather than on a circle. In a linear interval strip (G,A), G is a linear
interval graph and the cliques in A are made of contiguous vertices
at the end of the line segment. Fuzzy circular/linear interval graphs are
a slight generalization of circular/linear interval graphs.
The results considerably simplify for the subclass of quasi-line
graphs.

Theorem 7 ([8]). Let G(V,E) be a connected quasi-line graph. One
of the followings holds: G is a fuzzy circular interval graph; G is the com-
position of fuzzy linear interval strips.

We point out that while the two above results are not algorith-
mic, lighter versions of those have been recently algorithmized by
Hermelin, Mnich, van Leeuwen and Woeginger [30].
A different algorithmic decomposition theorem for claw-free
graphs was recently given by Faenza, Oriolo and Stauffer. From a
structural point of view, this result is much weaker than Theorem 6;
however, it is particularly useful when dealing with the MWSS prob-
lem, as we discuss in Section 4.1.

Theorem 8 ([19]). Let G(V,E) be a claw-free graph. In time
o(|(VI |El), one can find out whether oa(G) < 3, orG is almost nearly dis-
tance simplicial, or G is the composition of (|V|I) strips that are distance
simplicial strips or strips with stability number at most 3 and containing a
5-wheel (and provide the decomposition).

We denote by N(S), with S c V, the set of nodes of V \ S that
are adjacent to some node in S (we also use the notation N(v) for
N({ v)). A connected graph G is distance simplicial with respect to
a clique K if, for every j, a(Nj(K)) < 1, i.e., each neighborhood
Nj(K) of K is a clique; if there exists a vertex v such that G is
distance simplicial with respect to {v}, we simply say that G is dis-
tance simplicial. A distance simplicial strip is a strip (G, A), such that
G is distance simplicial with respect to each clique in A. A graph is
nearly distance simplicial if, for each v e V, G \ (N(v) u {v}) is dis-
tance simplicial. Almost nearly distance simplicial graphs are a slight
generalization. When G is quasi-line, Theorem 8 reads as follows:

Theorem 9 ([19]). Let G(V,E) be a quasi-line graph. In time
0(IVI IE|), one can find out whether G is the composition of 9(IVI)
distance simplicial strips, or G is almost nearly distance simplicial (and
provide the decomposition).


4 Following the Breakthrough: Stable Sets in Claw-Free
Graphs Revisited
4. I Faster Algorithm for Claw-Free Graphs
Suppose that we are interested in solving the MWSS problem on
a graph G that is the composition of some strips H1,...,Hk and
that, in particular, we are able to solve the same problem on each
strip. Building upon Lemma 3.1 we may reduce the former problem
to a matching problem. This goes as follows: we replace each strip
Hi, i 1,...k with suitable, simple, line strips H, i 1 ...,k, and






July 201 I


consider the graph G obtained by substituting HZ with Hz in the
composition. Following Lemma 3.1, G is a line graph, and therefore
a MWSS of G can be found by solving a matching problem. Finally,
from a MWSS of G we then recover a MWSS of G. We have in fact:

Theorem I 0 ([41 ]). The maximum weighted stable set problem on a
graph G that is the composition of some strips (G1, A1) ,..., (Gk, Ak)
can be solved in O(IV(G)12 log IV(G)| + Zi= ,...,k p(IV(Gi)))-time,
if each G' belongs to some class of graphs, where the same problem can
be solved in time O(pi (IV(Gt)I)).

Faenza, Oriolo and Stauffer [19] recently proposed a strongly
polynomial algorithm for solving the MWSS problem in a claw-free
graph G(V,E) that runs in O(IVI(IEI + |V|log|V|))-time, dras-
tically improving the previous best known complexity bound. This
algorithm builds upon Theorem 8, and, in the following, we sketch
how it deals with the different cases arising from that theorem. Let
G be a claw-free graph. If a(G) < 3, then a MWSS can be found by
enumeration. If G is the composition of strips, then the result fol-
lows from Theorem 10, as soon as we observe that we can find a
MWSS in a distance simplicial strip by dynamic programming, follow-
ing a construction and an algorithm from Pulleyblank and Shepherd
[48] for distance-claw-free graphs. The latter construction can be
used also when G is almost nearly distance simplicial. Without using
any sophisticated data structures, the algorithm can be implemented
asto run in O(IVI|(|E + |IV|log|V|))-time.

4.2 The Stable Set Polytope of Quasi-Line Graphs
When studying the polyhedral aspect of a composition of graphs, it
is standard to substitute some of the graphs with "gadgets" and to
derive the polyhedral description of the composition from the poly-
hedral descriptions of the composition of the simpler graphs [2, 3].
For the composition of strips, Chudnovsky and Seymour observed
[7, 53] that paths of length one or two were the appropriate gadgets
to prove the following:

Theorem I I ([7, 53]). The stable set polytope of a quasi-line graph
G that is not a fuzzy circular interval graph can be characterized by non-
negativity inequalities, clique inequalities and Edmonds' inequalities.

On one hand this theorem shows that the Ben Rebea conjecture
holds for such class of quasi-line graphs (Edmonds' inequalities are
particular clique family inequalities). On the other, with the help of
Theorem 7, it shows that all non-rank facet inducing inequalities for
quasi-line graphs appear in fuzzy circular interval graphs.
Eisenbrand, Oriolo, Stauffer and Ventura [17] were able to pro-
vide a linear description of STAB(G), when G is a fuzzy circular in-
terval graph. As fuzziness can be handled easily, in the following we
simply deal with circular interval graphs. For such graphs, let A be
the clique incidence matrix, when one restricts to cliques stemming
from the intervals. Then the stable set problem can be formulated
as: max { vcvxv : Ax < 1 ; xv e {0,1}, Vv e V What is
crucial is that the matrix A has the so-called circular one property,
i.e., there is an ordering of the columns such that, on each row, the
ones appear consecutively, under the convention that the first col-
umn is consecutive to the last. But then the linear relaxation P =
{x e Rn ( A)x < () } is such that Pk = P n {x : Xv k}
is integral for any integer k (using the equation v vxv k, the
system ( A,)x < (1) can be rewritten as a consecutive one system
and so pk is described by a totally unimodular system). This result
shows that the only missing inequalities are disjunctive cuts of the
form vv-, xv < k v Z-,,v xv > k + 1 from QSTAB(G). Careful
analysis of those disjunctive cuts allows one to prove that they are
clique family inequalities and therefore the Ben Rebea conjecture
holds true [17].


Theorem 12 ([ 17]). Conjecture I holds true.

4.3 The Stable Set Polytope of Claw-Free Graphs with a > 4 and No
Clique Cutsets
Galluccio, Gentile and Ventura [22] extended Theorem I I to deal
with the stable set polytope of a graph that is the composition
of arbitrary strips. Let G be the composition of a family of strips
H = (G, Ai),i = 1,.., k. Strips with only one extremity can be
easily handled because of a result of Chvatal [12]. Therefore assume
without loss of generality that each strip has two extremities. We
denote by G' the graph obtained from G' by adding a new node z
with N(z) A' u A', and by Giy the graph obtained from Gi by
adding two new nodes u and v such that N(u) A u {v} and
N(v) A u {u}. In [22] it is proved that the inequalities needed to
describe STAB(G) can be obtained by (appropriately) replacing the
inequalities defining STAB(GuV) and STAB(G]) in the stable set
polytope of a certain line graph G, derived from G by substituting
each HZ with a line strip.
In the following, we apply this result to claw-free graphs. Galluc-
cio, Gentile and Ventura [24, 25] managed to provide a descrip-
tion of the stable set polytope of the graphs G' and G'v associ-
ated with the strips (G', A') arising from Theorem 6: in particular,
they showed that non-negativity inequalities, rank inequalities, se-
quential lifting [43] of 5-wheel inequalities and sequential lifting of
geared inequalities are sufficient to describe both STAB(GQ) and
STAB(G',v). We have therefore:

Theorem I 3 ([25]). The stable set polytope of a claw-free graph with
stability number at least 4, non-fuzzy circular interval and with no clique-
cutset is defined by: non-negative inequalities, sequential lifting of multi-
ple geared inequalities, rank inequalities and sequential lifting of 5-wheel
inequalities. In particular, all inequalities are {0, 1, 2 -valued.

We recall that a description of rank inequalities in claw-free
graphs follows from the characterization of rank minimal facets in
[26]. As for (multiple) geared inequalities [21, 23], in claw-free graphs
they are {0, 1, 2 -valued facet defining inequalities that are "pro-
duced" from rank inequalities, by substituting one or multiple edges
with a gear, a graph that is made of two intertwined 5-wheels (see
Figure 3).
Combining Theorem 6, Theorem 12 and Theorem I 3, we have:

Theorem 14 ([25]). The stable set polytope of any claw-free graph
G without a clique-cutset and such that oa(G) > 4 is defined by: non-
negativity inequalities, clique-family inequalities, rank inequalities, sequen-
tial lifting of 5-wheel inequalities, and sequential lifting of multiple geared
inequalities.


x < 3


xo +2x. < 5 o + 2x. < 7


Figure 3. A rank, a geared and a multiple geared facet defining inequality


4.4 Extended Formulation and Separation for the Stable Set Polytope
of Claw-Free Graphs
Faenza, Oriolo and Stauffer [19] gave a characterization of
STAB(G), G claw-free, in an extended space. Their result builds
upon a suitable extended description of STAB(G) for a graph G






OPTIMA 86


that is the composition of a family of strips HZ = (G, A'), i 1,.., k.
In fact, Theorem 10 has a polyhedral interpretation in an extended
space. We proceed as in Section 4.1, and let G be the line graph
that arises by substituting each strip HZ with a suitable line strip HZ
in the composition. Given a polytope P = x e n1 : Ax < b} in
Rn, we call {(x, Ap) e n1 x R : Ax < Apb,Ap > 0} the homoge-
nized cone associated with P and the system Ax < Apb, Ap > 0 the
homogenization of the system Ax < b.

Theorem 15 ([18]). An extended formulation for STAB(G) is
(mainly) given by the homogenization of the (possibly extended) lin-
ear descriptions of STAB(GZ), STAB(Gi[V(G) \ UA zA]) and
STAB(Gi[V(G) \ A]) for all A e A' ; and STAB(G) where G is
a line graph with (9(k) vertices.

(The reader should rely on [18] to find out what is behind the
word "mainly" in the previous statement.) We now sketch how to
apply this construction to a claw-free graph G (V, E). By Theorem 8,
we know that in time 0(|V||IIEI) we can distinguish if a(G) < 3, if
G is almost nearly distance simplicial or if G is the composition of
distance simplicial strips and strips with stability number at most 3.
It is easy to write an extended formulation for a graph G with
small stability number. Indeed, let Xi,...,Xk be all the extreme
points of STAB(G) i.e., all stable set of size 0, 1, 2 or 3. The poly-
k k
tope Q (x,A) : x iAixi,A > 0, Ai = 1 is an ex-
tended formulation of STAB(G). Nearly distance simplicial graphs
are distance claw-free graphs, a class of graphs for which Pulleyblank
and Shepherd [48] gave a compact extended formulation based on
a dynamic programming algorithm for the MWSS problem (note that
this class also includes graphs that are distance simplicial graphs with
respect to some clique). We are left with the case where G is the
composition of a family of strips HZ (Gi, Ai), i 1,.., k. In this
case, building upon Theorem 15, we just need to show that we are
able to derive extended formulations for the stable set polytopes
associated with the strips. In fact, if either G' is a distance simplicial
graph with respect to some clique, or xo(Gi) < 3, then an extended
formulation for STAB(GZ) (or STAB(Gi[V(G)\ UA~ ,A]) etc.) fol-
lows from the above arguments.
We point out that the resulting extended formulation is simple
(a generalization of the union of polytopes [I]) and requires only
0(n) extra variables. Moreover, even though it might have expo-
nentially many Edmonds' inequalities, they are separable in polytime
[44]. Thus one can write an explicit linear formulation of the prob-
lem that could also be used as a strong relaxation for the variation
of the stable set problem in claw-free graph with additional side con-
straints. One should also observe that if there would exist a compact
extended description of the matching polytope, a well-known open
problem, then also this formulation would be compact.
Faenza, Oriolo and Stauffer [18] gave another extended formula-
tion of STAB(G), G claw-free, that is better suited for projection, as
it requires only one additional variable per strip. They derived from
the projection of this formulation on the original space an alterna-
tive view to Theorem 14 and, more important, a polytime separation
routine for STAB(G) (in the original space).

Theorem 16 ([ 18]). Let G (V, E) be a claw-free graph. It is possible to
separate in polynomial time over STAB(G) using only a separation rou-
tine for matching and the solution of 0(|V|) compact linear programs.


5 Open Questions
5.1 Complete Linear Description of STAB(G) in the Original Space
It follows from [12] and Theorem 14, that in order to provide a
linear description of the stable set polytope of any claw-free graph,


Figure 4. The complement of a claw-free graph G. The graph G induces the
facet: 2(x4 + x15 + x16) + 3(xi + x2 + x3 + x4) + 4(xs + x6 + x7 + xg +
x12) + 5(xg + x10 + x1 ) + 6(x13 + x17) < 8. Note that G is not quasi-line and
that c((G) = 3. (The picture is a courtesy of Tristram Bogart, Annie Raymond and
Rekha Thomas.)



we are left with characterizing the stable set polytopes of claw-free
graphs with stability number at most 3. In fact, following Theorem
12 we may restrict to claw-free, non-quasi-line graphs with stability
number at most 3. However, even as Cook [14, 52, 33] character-
ized the stable set polytope of any graph G with a(G) < 2, it seems
that characterizing the stable set polytopes of graphs with stability
number at most 3, is quite challenging, even if we restrict to claw-
free non-quasi-line graphs. In fact, we already pointed out in Section
2.2 that for claw-free, but non quasi-line, graphs with stability num-
ber 3 there exist facets with arbitrarily many coefficients [46]; see
Figure 4 for a facet inducing inequality with 5 different coefficients.
This is not the case for quasi-line graphs (see Theorem 12) or claw-
free graphs with stability number at least four and no clique-cutsets
(see Theorem I3).
Pecher and Wagler [45] worked on this question. While providing
some better understanding of affine independence for the remaining
difficult facets of SSP in claw-free graphs the so-called co-spanning
forest structure this work leaves the full characterization in the orig-
inal space still open. Indeed, Pecher and Wagler [45] do not provide
a 'proper' construction to produce a valid inequality associated with
a given structure besides, basically, exploiting the polar of the poly-
tope.
Observe that, in case one can solve the question above, the
essence of the complete linear description for claw-free graphs will
be significantly different from that of the stable set polytope of quasi-
line graphs, for which inequalities are defined algebraicallyy". This
suggests that, even if the case a(G) < 3 was solved, additional in-
sight might still be needed to get an elegant description of the stable
set polytope of claw free graphs, if any. The next question proposes
another standpoint on the problem that might lead to a simpler de-
scription of the polytope.

5.2 Calvillo's Theorem and the Intersection Property
Let STABk(G) : conv{x c {0, 11 : X, + xv < 1, V{u,v} e
E and Zv,,vxv = k }. Theorem 4 shows that STABk(G) =
STAB(G) n {x e RIV ~ vXv k} and it might suggest that
the stable set polytope of claw-free graphs has a nicer interpretation
when intersected with the hyperplanes {x e I1 I Yv x, = k}
for all integer k. This is indeed the case for quasi-line graphs. In
fact, building upon Theorem 4, Theorem 7, Theorem I I and some
arguments from [ 17], one can show that:

Lemma 5.1. Let G(V,E) be a quasi-line graph. For every integer k,
STABk (G) can be described by non-negativity inequalities, clique inequal-
ities, Edmonds' inequalities and Ycv x, k.






July 201 I


Can we hope for a similar result for claw-free graphs? Because
of 5-wheel structure, one can easily show that, in contrast to quasi-
line graphs, for claw-free graphs rank inequalities are not enough
to describe STABk(G); however a complete characterization of
STABk(G) might still be simple.

5.3 Minimal Linear Description for the Stable Set Polytope of
Quasi-Line Graphs
For the matching polytope, Edmonds and Pulleyblank [ 16] gave a de-
scription of the facets of the polytope, giving necessary and sufficient
conditions for an odd set of vertices to induce a facet. While the
Ben Rebea theorem gives a linear description of STAB(G) when G
is quasi-line, it does not provide necessary and sufficient conditions
for a family of cliques K and an integer p > 2 to induce a clique fam-
ily inequality that is facet inducing. The question is open but there
are a few results in this direction [26, 53, 54, 42].

5.4 The Chvatal-Gomory Rank of the Stable Set Polytope of
Quasi-Line Graphs
Sbihi [50] reported that Edmonds conjectured in 1973 that the sta-
ble set polytope of any claw-free graph G had Chvatal-Gomory rank
(CG-rank in the following) one from QSTAB(G). This was proven
false by Giles and Trotter [27], who provided a facet-defining in-
equality for STAB(G) with CG-rank two. The result was strength-
ened by Chvatal [I 3] who showed that there exist graphs with
x(G) = 2, and therefore claw-free, with CG-rank unbounded. In-
terestingly this construction does not extend to quasi-line graphs, as
building upon Theorem 12 and results in [14, 52, 33], one may show
that the CG-rank of a quasi line graph G with a(G) 2 is one.
However, facet-defining inequalities with CG-rank two exist also for
quasi-line graphs, as shown by Oriolo [40]. This raises the follow-
ing questions: is the CG-rank unbounded for quasi-line graphs or is
it bounded? (Actually, despite some efforts, we could not produce
for quasi-line graphs facet-defining inequalities with CG-rank bigger
than two). We mention that Pecher and Wagler [47] studied the
CG-rank of general clique family inequalities and gave some upper
bounds. Unfortunately they do not seem to be tight.

5.5 Improving the Complexity
The weighted matching problem in a graph H(W,F) can be solved
in O(IWI(I|WloglW + |F|) -time [20]. It follows that we can find
a MWSS in a line graph G(V,E) in time O(IV21loglVI). Follow-
ing the algorithm by Faenza, Oriolo and Stauffer presented in Sec-
tion 4.1 a MWSS in a claw-free graph G(V,E) can be found in time
O(I V1 (I V| log I V + |El)), i.e., slightly worse than for line graphs: can
we close this gap? We believe that this should be doable, in particular
for quasi-line graphs. Also note that the above algorithm uses only
elementary data structures, so one could try to lower its complexity
by using more sophisticated data structures.

5.6 A "Short" Proof of the Ben Rebea Theorem
One should note that Theorem 9 is quite close to Theorem 7. How-
ever, while the former theorem has a rather simple and direct proof,
the proof of the latter relies on the general structure of claw-free
graphs. We ask therefore whether it is possible to sharpen Theo-
rem 9 so as to prove the same characterization of Theorem 7. The
question of having a direct proof of Theorem 7 was raised already
by King [3 I1]. Note that such a proof, together with the proof of
Theorem I I and the proof that the Ben Rebea conjecture holds for
fuzzy circular interval graphs in [ 17], would provide a "short" proof
of Theorem 12.


Acknowledgements
We are grateful to Sam Burer, Yuri Faenza and Volker Kaibel for
comments on a draft of this article and to Tristram Bogart, Annie
Raymond and Rekha Thomas for allowing us to use Figure I and
Figure 4.

Gianpaolo Oriolo, Dipartimento di Informatica, Sistemi e Produzione, Uni-
versiti Tor Vergata, Roma. oriolo@disp.uniroma2.it
Gautier Stauffer, Bordeaux Institute of Mathematics, 351 course de la Libera-
tion, 33405 Talence, France. gautier.stauffer@math.u-bordeauxl.fr
Paolo Ventura, Istituto di Analisi dei Sistemi ed Informatica "A. Ruberti" del
CNR, Roma. paolo.ventura@iasi.cnr.it



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Discussion Column


Manfred W. Padberg

Node Packings in Graphs and Claw Free

Graphs


When Volker Kaibel called me a couple of weeks ago to ask me to
write a short historical note (as a discussion column to the article by
Gianpaolo Oriolo, Gautier Stauffer, and Paolo Ventura) about node
packing in graphs and especially in claw-free graphs, my reaction
was more or less: Boy, it has been something like forty years ago
that I've worked on that stuff! But I promised to do it and so here
it is. Most of what I have to say is about node packing in graphs
and the general context in which it happened, but some of it may
help to explain how the interest in claw-free graphs travelled from
Pittsburgh via Berlin, Bonn and New York to Rome.


I Beginnings
Had you asked me back in 1969/1970 about node packing or vertex
packing in graphs (bull-free, claw-free or whatever), you would have
drawn a blank stare: I would not have known what you were talking
about. These terms may have existed in graph theory, but not in inte-
ger programming. In integer programming we were concerned with
linear programs with practical applications: airline crew scheduling
problems, knapsack problems (with their roots in capital rationing
in finance) and traveling salesman problems (TSPs with their roots
in K. Menger's "Botenproblem" from the 1930's), to name just three
favorite problems of mine in those days. The intimate connections
between graph theoretical and integer or zero-one programming
problems that you young guys are all familiar with today just
had not been established yet. All right, to understand TSPs you need
a modicum of graph theory, but that's all. What got my interest in
this field were airline crew scheduling problems, perhaps because
in September 1968 I had left my native Westphalia in Germany to
fly to New York and then, by my ex-wife's car, to Pittsburgh, PA,
where I had obtained a Ford Foundation Fellowship to complete my
doctoral studies at Carnegie-Mellon University's GSIA and where I
specialized with Egon Balas, see my historical note "Mixed-integer Pro-
gramming 1968 and thereafter" in Annals of Operations Research,
2007, 149: 163-175. Despite Dantzig, Fulkerson and S. Johnson's
milestone 1954 paper Solution of a Large Scale Travelling Salesman Prob-
lem published in Operations Research a world record as it solved a
problem in 1,128 zero-one variables to optimality computational
integer programming had a bad name. Egon Balas' 1965 paper An Ad-
ditive Algorithm for Solving Linear Programming Programs with Zero-One
Variables published in Operations Research had bettered the picture
somewhat, but all of this was "overshadowed" by Ralph Gomory's
algebraic, some said "elegant", algorithm for integer programming
(first abstracted in 1958 and published in 1963), which computa-
tionally just did not work. Of the many references in the literature
to this effect, let me just mention Don Knuth's 1961 paper Minimiz-
ing Drum Latency Time in the Journal of the Association for Computing






July 201 I


Machinery and some articles in the book edited by Muth and Thomp-
son Industrial Scheduling, Prentice-Hall, 1963. In any case, the recent
"revival" of Gomory's mixed-integer cuts in computational integer
programming does not contradict what I'm saying because they are
based on "disjunctive" reasoning and thus different from the original
Gomory cuts developed in 1958 or so; see also my paper Classical
Cuts for Mixed-Integer Programming and Branch-and-Cut in Math. Meth.
Oper. Res. 2001, 53: 173-203 or its reprint in Annals of O.R. 2005,
139: 321-352. So much for the history as I found it back in 1968 in
Pittsburgh, PA.


2 Step I
By April/May of 1971 I had finished and defended my PhD thesis
Essays in Integer Programming at GSIA and prepared my return to
Germany, because of my obligation to do so under the conditions
of my Ford Foundation Fellowship. I wound up at the International
Institute of Management in Berlin with a three-year contract. On
March 25, 1970, Egon and I had submitted Chapter 2 of my thesis
On the Set-Covering Problem to Operations Research where it appeared
in the November-December 1972 issue. Chapter 3 of my thesis 'Sim-
ple' Zero-One Problems: Set Covering, Matchings and Coverings in Graphs
was widely distributed as the Management Sciences Report No. 235 of
CMU's GSIA and submitted to Mathematical Programming sometime
in late 1971. In it I had abstracted from Egon's and mine results of
Chapter 2 a property of the said problems called 'Simplicity' of a
polytope (which I myself found in early 1972 to be erroneous).
In any case, my Chapter 3 contained the first results on the facets
of these polytopes, namely the clique and 'lifted' odd-cycle facets.
The pertaining correct results of it were published in somewhat im-
proved form in On the Facial Structure of Set Packing Polyhedra, Mathe-
matical Programming, 1973, 5: 199-215. Just for completeness, Chap-
ter 4 of my thesis Equivalent Knapsack-type Formulations of Bounded
Integer Linear Programs: An Alternative Approach appeared in Naval Re-
search Logistics Quarterly, 1972, 19: 699-708 and one of the technical
appendices A Remark on "An Inequality for the Number of Lattice Points
in a Simplex" in SIAM journal of Applied Math., 1971, 20: 638-641.
Another technical appendix of my thesis contained some results on
"adjacent vertices cuts", but these were just again "cuts" and not
"facet-defining cutting-planes" and thus I never published that stuff.
Voila, that was essentially the content of my 1971 thesis. So much
for those who still recently asked themselves what my thesis was all
about. Just read the published stuff.


3 Step II
I had wanted to test facet-defining cutting-planes for node and set
packing problems computationally in Berlin, but there were just no
adequate computing facilities in Berlin and also a lack of test prob-
lems. Karla Hoffman and I did so eventually in our paper Solving Airline
Crew Scheduling Problems by Branch-and-Cut published in Management
Science, 1993, 39: 657-682. Needless to say, it worked wonderfully,
but here I am jumping way ahead of time.
I landed a job at New York University's Graduate School of Busi-
ness as of September 1974 when my obligation to the Ford Foun-
dation to stay in Germany was over. On my way from Berlin to
New York I "stopped" for about half a year at Bernhard Korte's
then new Institute for Operations Research and Econometrics at
Bonn University, where I met Martin Groetschel. I don't think that
it is necessary to recall our joint work on the traveling salesman
polytope here. Besides the theoretical work, Martin's 1976 thesis
contained the solution to optimality of a 120-city TSP using only
facet-defining cutting-planes that's a linear program in 7,140 zero-
one variables and thus another world record! Wunderbar, because


that is exactly what I had in mind when I started out in 1970 to
look for facet-defining cutting-planes, rather than arbitrary "cuts"
with no proven mathematical properties other than "validity" for
the problem in question. I won't recall either in detail my compu-
tational work on the TSP (with S. Hong, then with H. Crowder
and later with Giovanni Rinaldi) as well as on other problems pur-
sued with the same goal to prove "empirically" the value of facet-
defining cutting-planes in actual computation. But I will recall my
work with M. Ram Rao Odd Minimum Cut-Sets and b-Matchings, Math-
ematics of Operations Research, 1982, 7: 67-80, which we presented
at the 1979 Mathematical Programming Symposium in Montreal. For
in the meantime, the late Leonid Khachian had proved that linear
programming problems could be solved in polynomial time by the
ellipsoid method and his result had traveled to the West just around
1979. After my presentation, Jack Edmonds and Laslo Lovasz con-
jectured that Ram and I had just given another poly-time algorithm
for b-matchings in graphs. This turned out to be true and Ram and
I were delighted when Martin Groetschel and Olaf Holland showed
in A cutting-plane algorithm for minimum perfect 2-matching, Computing,
1987, 39: 32 7-344, that our pure cutting-plane algorithm (using the
simplex method, of course, instead of the ellipsoid algorithm) out-
performed Edmonds' graphical algorithm in practical computation. I
should add that purely graphical problems occur only rarely in prac-
tice and are frequently complicated by additional constraints such as
capacity and/or capital constraints which necessitates a cutting-plane
approach.
The major consequence of Khachian's ellipsoid method was the
equivalence of optimization and separation in terms of poly-time solv-
ability, a result that was obtained in early 1980 independently by
three different groups of researchers: Groetschel, Lovasz and Schri-
jver, Karp and Papadimitriou and Padberg and Rao. (You'll find a
proof of this equivalence, e.g., in Chapter 9 of my book Linear Op-
timization and Extensions, 1995, 2nd ed. 1999, Springer Verlag). This
equivalence generalized, of course, Edmonds and Lovasz's conjec-
ture mentioned above. But it also reinforced the "hunt" for facet-
defining cutting-planes that had followed my initial findings that facets
of the convex hull of the integer solutions could indeed be found
for (some) integer and mixed-integer programs, like node covering,
node packing, set packing problems and then knapsack problems,
TSPs, etc.



4 Step III
Also in or around 1979 I learned that the late George Minty had
generalized Edmonds' matching algorithm and found a poly-time al-
gorithm for vertex packing in claw-free graphs, see journal of Com-
binatorial Theory B, 1980, 284-304, and independently of him Najiba
Sbihi, see Algorithme de Recherche d'un Stable de Cardinalite Maximum
dans un Graphe sans Etoile, Discrete Mathematics, 1980, 29: 53-76,
as well. Once the equivalence of optimization and separation had
been established, given the poly-time solvability of weighted vertex
packing in K1,3-free graphs, the separation problem for the associ-
ated convex hull of node packing in such graphs had to be solvable
in poly-time as well. Being an eternal optimist, I put the problem of
finding a complete minimal linear description for this problem on my
list of things to do, but never came around to attacking this problem
alone.
Sometime in 1981/1982, the late Mario Lucertini of Rome's Uni-
versita Tor Vergata invited me to do a two-week intensive course on
combinatorial optimization in Rome it must have been in August
of 1982, because it was awfully hot and the class room had no air-
conditionning. I met through Lucertini Giovanni Rinaldi and Antonio
Sassano, who had just re-joined the Italian CNR after having worked






OPTIMA 86


for a while in their own company. After one of my courses, the four
of us discussed over a cool beer in one of the shady squares of
Rome a possible visit of Antonio and Giovanni with me at New York
University. Antonio Sassano came in 1983/1984 to work with me at
NYU, Giovanni came a little while later. I suggested to Antonio to
work on the facial structure of the polytope of vertex packing in
K1,3-free graphs, but perhaps due to the relatively short time that
Antonio stayed with me in New York, we did not get enough sub-
stantial results on the problem to write a joint paper on it. When
he returned to Rome, he took the problem along with him and the
desire to solve it; just look at Sassano's homepage at the Universita
La Sapienza, where several papers on this topic (with various coau-
thors) are listed. I am sure that Antonio Sassano has a lot to do with
the progress made in Italy on this problem like the new algorithm by
Faenza, Oriolo, and Stauffer and the new polyhedral results by Gal-
luccio, Gentile, and Ventura. Personnally, what I find very interesting
is, of course, a pure cutting-plane algorithm for this problem, like
the one that Ram Rao and I found for b-matchings in graphs and that
(simplex method or ellipsoid method, I don't care) is computationally
efficient and permits other complicating constraints to be added. I
am happy to hear that the separation routine via the new extended
formulation due to Faenza, Oriolo, and Stauffer provides this.

Manfred W. Padberg, Emeritus Professor, New York University, Stern School
of Business, 17 rue Vend6me 13007 Marseille, France
manfred4@wanadoo.fr






Obituary


Oleg Burdakov, John Dennis, and Jorge More

Charles G. Broyden, 1933-20 I1

Charles George Broyden was born
February 2nd, 1933, in England. He
received his degree in Physics from
rKings College London in 1955. He
spent the first ten years of his ca-
reer in industry. In 1967, he moved
to the University of Essex where
N he became a professor and, later,
dean of the School of Mathemat-
ics. In 1986, he decided to retire
early to become a traveling scholar,
but in 1990, he accepted an ap-
pointment as a professor of nu-
merical analysis at the University of
Charles George Broyden, Bologna.
Sweden, 2002 (Photo: Charles received international
leg Burrdakov) recognition for his seminal 1965

paper in Mathematics of Computa-
tion, in which he proposed two methods for solving systems of equa-
tions. They later became known as Broyden's methods. Another of
his most important achievements was the derivation of the Broyden-
Fletcher-Goldfarb-Shanno (BFGS) updating formula, one of the key
tools used in optimization. Moreover, Charles was among those who
derived the symmetric rank-one updating formula, and his name is
also attributed to the Broyden family of quasi-Newton methods.
At Bologna, Charles shifted the focus of his research to numerical
linear algebra and, in particular, to conjugate gradient methods and


to the taxonomy of these methods. Some of the main results of that
period are summarized in his 2004 book with M.T. Vespucci Krylov
Solvers for Linear Algebraic Systems.
In recognition of his fundamental contributions to the develop-
ment of optimization and numerical mathematics, the journal Opti-
mization Methods and Software (OMS) established the Charles Broy-
den prize. It is awarded yearly for the best paper published in
OMS.
When Charles learned about the prize, he modestly noted that
"I discovered my algorithms because I was in the right place at the
right time". Being in the right place at the right time once could be
good luck, but if this happens several times, this clearly indicates tal-
ent. Indeed, one can hardly find a book on numerical optimization
where the discoveries of Charles Broyden are not mentioned.
Charles Broyden died on May 20, 201 I. We will remember him as
a highly dedicated, modest, and honest researcher, respected by his
many friends and collaborators around the world. We express our
sympathy to his wife Joan, his children and grandchildren.








Announcements


Call for Nominations for the 2012

Beale-Orchard-Hays Prize

Nominations are invited for the 2012 Beale-Orchard-Hays Prize for
excellence in computational mathematical programming that will be
awarded at the International Symposium on Mathematical Program-
ming to be held in Berlin in August 2012.
The Prize is sponsored by the Mathematical Optimization Society,
in memory of Martin Beale and William Orchard-Hays, pioneers in
computational mathematical programming. Nominated works must
have been published between Jan I, 2009 and Dec 31, 2011, and
demonstrate excellence in any aspect of computational mathemat-
ical programming. "Computational mathematical programming" in-
cludes the development of high-quality mathematical programming
algorithms and software, the experimental evaluation of mathemat-
ical programming algorithms, and the development of new methods
for the empirical testing of mathematical programming techniques.
Full details of prize rules and eligibility requirements can be found at
http://www.mathopt.org/?nav=boh.
The 2012 Prize Committee consists of Michael Ferris (Chair),
Philip Gill, Tim Kelley, and Jon Lee.
Nominations can be submitted electronically or in writing, and
should include detailed publication details of the nominated work.
Electronic submissions should include an attachment with the final
published version of the nominated work. If done in writing, submis-
sions should include four copies of the nominated work. Supporting
justification and any supplementary material are strongly encouraged
but not mandatory. The Prize Committee reserves the right to re-
quest further supporting material and justification from the nomi-
nees.
Nominations should be submitted to:
Prof. Michael Ferris, Computer Sciences Department, University
of Wisconsin, 1210 West Dayton Street, Madison, WI 53706, USA
Email: ferris@cs.wisc.edu
The deadline for receipt of nominations is January 15, 2012.






July 201 I


ISMP 2012 in Berlin

The 2 Ist International Symposium on Mathematical Programming (ISMP
2012) will take place in Berlin, Germany, August 19-24, 2012. ISMP
is a scientific meeting held every three years on behalf of the Math-
ematical Optimization Society (MOS). It is the world congress of
mathematical optimization where scientists as well as industrial users
of mathematical optimization meet in order to present the most re-
cent developments and results and to discuss new challenges from
theory and practice.


Conference Topics
The conference topics address all theoretical, computational and
practical aspects of mathematical optimization including:
o integer, linear, nonlinear, semidefinite, conic and constrained pro-
gramming
o discrete and combinatorial optimization
o matroids, graphs, game theory, network optimization
o nonsmooth, convex, robust, stochastic, PDE-constrained and
global optimization
o variational analysis, complementarity and variational inequalities
o sparse, derivative-free and simulation-based optimization
o implementations and software
o operations research
o logistics, traffic and transportation, telecommunications, energy
systems, finance and economics


Conference Venue
The Symposium will take place at the main building of TU Berlin in
the heart of the city close to the Tiergarten park.
The opening ceremony will take place on Sunday, August 19, 2012,
at the Konzerthaus on the historic Gendarmenmarkt which is con-
sidered one of the most beautiful squares in Europe. The opening
session will feature the presentation of awards by the Mathemati-
cal Optimization Society accompanied by symphonic music. This is
followed by the welcome reception with a magnificent view on Gen-
darmenmarkt.
The conference dinner will take place at the Haus der Kulturen
der Welt ("House of the Cultures of the World") located in the
Tiergarten park with a beer garden on the shores of the Spree river
and a view on the German Chancellery.


NO

bl I LIH



Registration and Important Dates
The conference registration will open before December 201 I1. The
abstract submission deadline will be April 15, 2012, the early regis-
tration deadline June 15, 2012.
In accordance with the new MOS membership fees policy, ISMP
2012 will offer three early registration rates for regular attendees
(not students, not retired, and not lifetime members of MOS):
o Euro 340 including MOS membership for 2013
o Euro 390 including MOS membership for 2013 and 2014
o Euro 415 including MOS membership for 2013-2015
The early registration rates for retirees (not lifetime members of
MOS) are
o Euro 190 including MOS membership for 2013
o Euro 215 including MOS membership for 2013 and 2014
o Euro 230 including MOS membership for 2013-2015
The early registration rate for students is Euro 160. The early reg-
istration rate for lifetime members of MOS is Euro 280. The regis-
tration rates for late registration (after June 15, 2012) will be higher
(details to be announced).

Webpage etc.
More details (including registration rates, hotel prices, sponsorship
opportunities, exhibits etc.) can be found on the conference web
pages at www.ismp20 2.org.


...... If,.


Haus der Kulturen der Welt ("House of the Cultures of the World") located in the
Tiergarten park (Photo: Christoph Eyrich)


Main building of TU Berlin in the heart of the city close to the Tiergarten park
(Photo: TU Berlin/Dahl)






OPTIMA 86


Call for Site Pre-Proposals

ISMP 2015

The Symposium Advisory Committee of the Mathematical Opti-
mization Society issues a call for pre-proposals to organize and host
ISMP 2015, the triennial International Symposium on Mathematical
Programming.
ISMP is the flagship event of our society, regularly gathering over
a thousand scientists from around the world. The conference will be
held in or around the month of August, 2015. Hosting ISMP provides
a vital service to the mathematical optimization community and of-
ten has a lasting effect on the visibility of the hosting institution. It
also presents a significant challenge. This call for pre-proposals is
addressed at local groups willing to take up that challenge.
Preliminary bids will be examined by the Symposium Advisory
Committee (SAC), which will then issue invitations for detailed bids.
The final decision will be made and announced during ISMP 2012 in
Berlin. Member of the SAC are
Jeff Linderoth, Chair linderoth@wisc.edu,
Shabbir Ahmed, sahmed@isye.gatech.edu,
Kurt Anstreicher, kurt-anstreicher@uiowa.edu,
Anders Forsgren, andersf@kth.se,
Martine Labbe, mlabbe@ulb.ac.be,
Rudiger Schultz ruediger.schultz@uni-due.de, and
Shuzhong Zhang, zhangs@umn.edu.
Preliminary bids should be brief and contain information pertain-
ing to the
(1) Location,
(2) Facilities,
(3) Logistics: Accommodation and transportation, and
(4) Likely local organizers.
Further information can be obtained from any member of the
advisory committee. Please address your preliminary bids until
September 15, 2011 to Jeff Linderoth linderoth@wisc.edu.


Conic Optimization Special Issue for

Pacific Journal of OPTIMIZATION

Original research articles are solicited for a forthcoming issue of
Pacific Journal of OPTIMIZATION dedicated to Conic Optimization.
Potential articles may focus on (or relate to) any subset of the
theoretical, computational and practical aspects of Conic Optimiza-
tion.
Over the last two decades, there has been a significant advance
in the research of interior point methods and conic optimization.
Today conic optimization has emerged as a major computational
paradigm and has made significant impact on numerous problems
previously considered intractable or difficult to approximate. This
special issue aims to solicit papers that provide original research on
conic optimization theory and algorithms, including available soft-
ware. Also welcome are papers that deal with emerging, meaning-
ful applications; or that give friendly overviews of certain theoret-
ically advanced conic optimization techniques relevant to applica-
tions.
Examples of topics that will be addressed in this special issue in-
clude, but are not limited to:
- Theory and algorithms for large scale conic optimization
- Computational study of conic optimization algorithms
- Relaxation techniques based on conic optimization
- Applications of conic optimization in science and engineering
(e.g., image processing, compressed sensing, digital communica-
tions and networks)
Instructions for authors are available on the web:
http://www.ybook.co.jp/online/forauthors.htm
Please e-mail the pdf-file of your full manuscript with a cover mes-
sage in plain text to one of the guest editors by September I, 2011.
Zhi-Quan Luo, University of Minnesota, luozq@umn.edu
Levent Tunnel, University of Waterloo, ltuncel@math.uwaterloo.ca
Naihua Xiu, Beijing Jiaotong University, nhxiu@center.njtu.edu.cn


IMPRINT
Editor: Katya Scheinberg, Department of Industrial and Systems Engineering, Lehigh University, Bethlehem, PA, USA. katyascheinberg@gmail.com Co-Editors:
Samuel Burer, Department of Management Sciences, The University of Iowa, Iowa City, IA 52242-1994, USA. samuel-burer@uiowa.edu m Volker Kaibel, Institut
fur Mathematische Optimierung, Otto-von-Guericke Universitat Magdeburg, Universitatsplatz 2, 39108 Magdeburg, Germany. kaibel@ovgu.de Founding Editor:
Donald W. Hearn Published by the Mathematical Optimization Society. m Design and typesetting by Christoph Eyrich, Berlin, Germany. optima@0x45.de .
Printed by Oktoberdruck AG, Berlin, Germany.


Application for Membership
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and not for the benefit of any library or institution.
D I will pay my membership dues on receipt of your invoice.
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Cheques or money orders should be made
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