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2 OPTIMA85 VolkerKaibelExtendedFormulationsinCombinatorialOptimization1Introduction LinearProgrammingbasedmethodsandpolyhedraltheoryfor mthe backboneoflargepartsofCombinatorialOptimization.The basic paradigmhereistoidentifythefeasiblesolutionstoagive nproblemwithsomevectorsinsuchawaythattheoptimizationprob lem becomestheproblemofoptimizingalinearfunctionoverthe niteset X ofthesevectors.Theoptimalvalueofalinearfunction over X isequaltoitsoptimalvalueovertheconvexhull conv X  f P x 2 X x x : P x 2 X x  1 ; O g of X .AccordingtotheWeyl MinkowskiTheorem[ 33 25 ],every polytope (i.e.,theconvexhullof anitesetofvectors)canbewrittenasthesetofsolutionst oa systemoflinearequationsandinequalities.Thusoneendsu pwitha linearprogrammingproblem. Asforthemaybemostclassicalexample,letusconsiderthes et M n ofallmatchingsinthecompletegraph K n  V n ;E n  on n nodes(whereamatchingisasubsetofedgesnotwoofwhichsha re acommonend-node).Identifyingeverymatching M E n withits characteristicvector M 2f 0 ; 1 g E n (where M e  1 ifandonly if e 2 M ),weobtainthe matchingpolytope P match n  conv f M : M 2M n g .Inoneofhisseminalpapers,Edmonds[ 13 ]proved that P match n equalsthesetofall x 2 R E n  thatsatisfytheinequalities xv 1 forall v 2 V n and xE n S bj S j = 2 c forall subsets S V n of odd cardinality 3 j S j n (where v isthe setofalledgesincidentto v E n S isthesetofalledgeswithboth end-nodesin S ,and xF  P e 2 F x e ).Noinequalityinthissystem, whosesizeisexponentialin n ,isredundant. Thesituationisquitesimilarforthe permutahedron P perm n i.e.,theconvexhullofallvectorsthatarisefrompermutin gthe componentsof  1 ; 2 ;:::;n .Rado[ 29 ]provedthat P perm n is describedbytheequation xn  nn  1 = 2 andtheinequalities xS j S j  j S j 1 = 2 forall  6 S n (with n f 1 ;:::;n g ),noneofthe 2 n 2 inequalitiesbeingredundant.Howeverifforeachpermutation : n n weconsiderthecorresponding permutationmatrix y 2f 0 ; 1 g n n (satisfying y ij  1 ifandonlyif i  j )ratherthanthevector  1 ;:::;n ,weobtainamuchsmallerdescriptionof theresultingpolytope,since,accordingtoBirkhoff[ 7 ]andvon Neumann[ 32 ],theconvexhull P birk n (the Birkhoff-Polytope )of all n n -permutationmatricesisequaltothesetofall doublystochastic n n -matrices(i.e.,nonnegative n n -matricesallof whoserow-andcolumnsumsareequaltoone).Itiseasytoseethatthepermutahedron P perm n isalinearprojectionof theBirkhoff-polytope P birk n viathemapdenedby py i  P nj  1 jy ij .Since,foreverylinearobjectivefunctionvector c 2 R n wehave max fh c;x i : x 2 P perm n g max f P ni  1 P nj  1 jc i y ij : y 2 P birk n g ,onecanuse P birk n (thatcanbedescribedby n 2 nonnegativityinequalities)insteadof P perm n (whosedescription requires 2 n 2 inequalities)withrespecttolinearprogrammingrelatedissues. Ingeneral,an extension ofapolytope P R n isapolyhedron Q R d (i.e.,anintersectionofnitelymanyafnehyperplanesandhalfspaces)togetherwithalinearprojection p : R d R n satisfying P  pQ .Anydescriptionof Q bylinearequationsand linearinequalitiesthen(togetherwith p )isan extendedformulation of P .The size oftheextendedformulationisthenumberofinequalitiesinthedescription.Notethatweneitheraccount forthe numberofequations(wecangetridofthembyeliminatingvar iables)norforthenumberofvariables(wecanensurethatthe re arenotmorevariablesthaninequalitiesbyprojecting Q totheorthogonalcomplementofits linealityspace ,wherethelatteristhe spaceofalldirectionsoflinescontainedin Q ).If T 2 R n d is thematrixwith py  Ty ,then,forevery c 2 R n ,wehave max fh c;x i : x 2 P g max fh T t c;y i : y 2 Q g Intheexampledescribedabove, P birk n thusprovidesanextendedformulationof P perm n ofsize n 2 .Itisnotknownwhether onecandosomethingsimilarforthematchingpolytopes P match n (wewillbebacktothisquestioninSection4.2).Howeverthe re aremanyotherexamplesofniceandsmallextendedformulati ons forpolytopesassociatedwithcombinatorialoptimization problems. Theaimofthisarticleistoshowafewofthemandtoshedsomelightonthegeometric,combinatorialandalgebraicbackgr oundof thisconceptthatrecentlyhasreceivedincreasedattentio n.Thepresentationisnotmeanttobeasurvey(forthispurpose,weref erto VanderbeckandWolsey[ 31 ]aswellastoCornujols,Conforti, andZambelli[ 11 ])butratheranappetizerforinvestigatingalternativepossibilitiestoexpresscombinatorialoptimization problemsby meansoflinearprograms. Whilewewillnotbeconcernedwithpracticalaspectshere,e xtendedformulationshavealsoproventobeusefulincomputa tions. YouwillndmoreonthisinLaurenceWolsey'sdiscussioncol umn below.Fundamentalworkwithrespecttounderstandingthec onceptofextendedformulationsanditslimitshasbeendoneby Mihalis Yannakakisinhis1991-paper ExpressingCombinatorialOptimization ProblemsbyLinearPrograms [ 34 ](seeSect.3.3and4).Wearevery happythatheshareswithussomeofhisthoughtsonthesubjec tin anotherdiscussioncolumn. 2SomeExamples2.1SpanningTrees The spanningtreepolytope P spt n associatedwiththecomplete graph K n  V n ;E n  on n nodesistheconvexhullofallcharacteristicvectorsofspanningtrees,i.e.,ofallsubsetsofedge sthatform connectedandcycle-freesubgraphs.Inanotherseminalpap er,Edmonds[ 14 ]provedthat P spt n isthesetofall x 2 R E n  thatsatisfy theequation xE n   n 1 andtheinequalities xE n S j S j 1 forall S V n with 2 j S j
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April2011 3 thatequationsarewritten,e.g.,aspairsofinequalities) .Thenthe system A i z i i b i forall i 2 q P qi  1 i  1 O withvariables z i 2 R n forall i 2 q and 2 R q isanextendedformulation for P ofsize f 1  f q  q ,wheretheprojectionisgivenby z 1 ;:::;z q ; z 1  z q .ThishasbeenprovedrstbyBalas (see,e.g.,[ 3 ]),evenforpolyhedrathatarenotnecessarilypolytopes (whereinthisgeneralcase P needstobedenedasthetopological closureoftheconvexhulloftheunion). 2.3DynamicProgramming Whenacombinatorialoptimizationproblemcanbesolvedbya dynamicprogrammingalgorithm,oneoftencanderiveanextend edformulationfortheassociatedpolytopewhosesizeisroughlyb ounded bytherunningtimeofthealgorithm. Asimpleexampleisthe0/1-Knapsackproblem,whereweare givenanonnegativeintegralweightvector w 2 N n ,aweight bound W 2 N ,andaprotvector c 2 R n ,andthetaskisto solve max fh c;x i : x 2 Fw;W g with Fw;W f x 2f 0 ; 1 g n : h w;x i W g .Aclassicaldynamicprogrammingalgorithmworksby settingupanacyclicdirectedgraphwithnodes s   0 ; 0  t ,and i;! forall i 2 n 2f 0 ; 1 ;:::;W g andarcsfrom i;! to i 0 ;! 0  ifandonlyif ia j s sortseverysequence a 1 ;:::;a n  2 R intonon-decreasingorder. TheconstructionprincipleofGoemanshasbeengeneralized tothe frameworkof reectionrelations [ 21 ],which,forinstance,canbe usedtoobtainsmallextendedformulationsforall G -permutahedra ofnitereectiongroups G (see,e.g.,Humphreys[ 19 ]),includingextendedformulationsofsize O  log m ofregular m -gons,previously constructedbyBen-TalandNemirovski[ 6 ].Anotherapplicationof reectionrelationsyieldsextendedformulationsofsize O n log n for Huffman-polytopes ,i.e.,theconvexhullsoftheleaves-to-rootdistancesvectorsinrootedbinarytreeswith n labelledleaves.Note thatlineardescriptionsofthesepolytopesintheoriginal spacesare verylarge,rathercomplicated,andunknown(seeNguyen,Ng uyen, andMaurras[ 26 ]). Thelistofcombinatorialproblemsforwhichsmall(andnice ) extendedformulationshavebeenfoundcomprisesmanyother s, amongthemperfectmatchingpolytopesofplanargraphs(Bar ahona[ 5 ]),perfectlymatchablesubgraphpolytopesofbipartite graphs(BalasandPulleyblank[ 4 ]),stable-setpolytopesofdistanceclaw-freegraphs(PulleyblankandShepherd[ 28 ]),packing andpartitioningorbitopes[ 15 ],subtour-eliminationpolytopes(Yannakakis[ 34 ]and,forplanargraphs,Rivin[ 30 ],Cheung[ 9 ]),andcertainmixed-integerprograms(see,e.g.,Conforti,diSumma ,Eisenbrand,andWolsey[ 12 ]). 3Combinatorial,Geometric,andAlgebraic Background 3.1FaceLattices Anyintersectionofapolyhedron P withtheboundaryhyperplane ofsomeafnehalfspacecontaining P iscalleda face of P .Theempty setand P itselfareconsideredtobe(non-proper)facesof P as well.Theproperfacesofathree-dimensionalpolytopethus areits vertices,edges,andthepolygonsthatmakeuptheboundaryo f P Partiallyorderedbyinclusion,thefacesofapolyhedron P forma lattice L P ,the facelattice of P .Theproperfacesthataremaximal withrespecttoinclusionarethe facets of P .Equivalently,thefacets of P arethosefaceswhosedimensionisonelessthanthedimensionof P .Anirredundantlineardescriptionof P hasexactlyone inequalityforeachfacetof P If Q R d isanextensionofthepolytope P R n withalinearprojection p : R d R n ,thenmappingeachfaceof P toits preimagein Q under p denesanembeddingof L P into L Q Figure1illustratesthisembeddingforthetrivialextensi on Q  f y 2 R V : P x 2 X y x  1 g of P  conv X via py  P x 2 X y x x for X fe1 ; e1 ;:::;e4 ; e4 g (thus P isthe cross-polytope in R 4 with 16 facetsand Q isthe standard-simplex in R 8 with 8 facets).As thisguresuggests,constructingasmallextendedformula tionfora polytope P meanstohidethefacetsof P inthefatmiddlepartof thefacelatticeofanextensionwithfewfacets.

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4 OPTIMA85 Figure1. Embeddingofthefacelatticeofthe 4 -dimensionalcross-polytopeintothefacelatticeofthe 7 -dimensionalsimplex 3.2SlackRepresentations Let P f x 2A : Ax b g R n beapolytopewithafnehull A a P A 2 R m n ,and b 2 R m .Theafnemap : A! R m with 'x  b Ax (the slackmap of P w.r.t. Ax b )isinjective.We denoteitsinverse(the inverseslackmap )onitsimage,theafnesubspace A ' A  R m ,by : A!A .Thepolytope P  A\ R m the slack-representation of P w.r.t. Ax b ,isisomorphicto P with 'P  P and ' P  P If Z R m isanitesetofnonnegativevectorswhose convexconic hull ccone Z f P z 2 Z z z : O g R m contains P  A\ R m thenwehave P  A \ ccone Z ,andthus,thesystem P z 2 Z z z 2 A and z 0 (forall z 2 Z )providesanextendedformulationof P ofsize j Z j viatheprojection ' P z 2 Z z z .Letuscallsuchan extensiona slackextension andtheset Z a slackgeneratingset of P (bothw.r.t. Ax b ). Nowsupposeconverselythatwehaveanyextendedformulatio n of P ofsize q deninganextension Q thatis pointed (i.e.,thepolyhedron Q doesnotcontainaline).Asforpolytopesabove(which inparticulararepointedpolyhedra),wecanconsideraslac krepresentation Q R q of Q andthecorrespondinginverseslackmap Thenwehave 'p  Q  P ,where p istheprojectionmapof theextension.Ifthesystem Ax b is binding for P ,i.e.,eachof itsinequalitiesissatisedatequationbysomepointfrom P ,then onecanshow(byusingstrongLP-duality)thatthereisa nonnegative matrix T 2 R m q  with 'p  z  T z forall z 2 Q ,thus P  T Q .Hencethecolumnsof T formaslackgeneratingsetof P

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April2011 5 (w.r.t. Ax b ),yieldingaslackextensionofsize q .Aseverynonpointedextensionofapolytopecanbeturnedintoapointedo ne ofthesamesizebyprojectiontotheorthogonalcomplemento f thelinealityspace,weobtainthefollowingresult,wheret he extensioncomplexity ofapolytope P isthesmallestsizeofanyextended formulationof P Theorem1 ([ 17 ]) Theextensioncomplexityofapolytope P isequal totheminimumsizeofallslackextensionsof P Aseveryslackextensionofapolytopeisbounded(andsincea ll boundedpolyhedraarepolytopes),Theorem1impliesthatth eextensioncomplexityofapolytopeisattainedbyanextension that isapolytopeitself.Furthermore,inTheorem1onemaytaket he minimumovertheslackextensionsw.r.t.anyxedbindingsy stemof inequalitiesdescribing P .Inparticular,alltheseminimaconcide. 3.3NonnegativeRank Nowlet P  conv X f x 2 a P : Ax b g R n beapolytope withsomeniteset X R n and A 2 R m n b 2 R m .The slackmatrix of P w.r.t. X and Ax b is 2 R m X  with i;x  b h A i;? ;x i Thustheslackrepresentation P R m of P (w.r.t. Ax b )isthe convexhullofthecolumnsof .Consequently,ifthecolumnsofa nonnegativematrix T R m f  formaslackgeneratingsetof P thenthereisanonnegativematrix S 2 R f X  with  TS .Conversely,foreveryfactorization  T 0 S 0 oftheslackmatrixinto nonnegativematrices T 0 2 R m f 0   and S 0 2 R f 0  X  ,thecolumns of T 0 formaslackgeneratingsetfor P Thereforeconstructinganextendedformulationofsize f for P amountstondingafactorizationoftheslackmatrix  TS into nonnegativematrices T with f columnsand S with f rows.Inparticular,wehavederivedthefollowingresultthatessentia llyisdueto Yannakakis[ 34 ](seealso[ 17 ]).Here,the nonnegativerank ofamatrixistheminumumnumber f suchthatthematrixcanbewritten asaproductoftwononnegativematrices,wheretherstoneh as f columnsandthesecondonehas f rows. Theorem2. Theextensioncomplexityofapolytope P isequaltothe nonnegativerankofitsslackmatrix(w.r.t.anyset X andbindingsystem Ax b with P  conv X f x 2 a P : Ax b g ). Clearly,thenonnegativerankofamatrixisboundedfrombel ow byitsusualrankasknownfromLinearAlgebra.Thereisalsoq uite someinterestinthe nonnegative rankof(notnecessarilyslack)matricesingeneral(see,e.g.,CohenandRothblum[ 10 ]). 4FundamentalLimits4.1GeneralLowerBounds Everyextension Q ofapolytope P hasatleastasmanyfacesas P ,as thefacelatticeof P canbeembeddedintothefacelatticeof Q (see Sect.3.1).Sinceeachfaceistheintersectionofsomefacet s,one ndsthattheextensioncomplexityofapolyhedronwith facesis atleast log .ThisobservationhasrstbeenmadebyGoemans[ 18 ] inordertoarguethattheextensioncomplexityofthepermut ahedron P perm n isatleast n log n Supposethat  TS isafactorizationofaslackmatrix of thepolytope P intononnegativematrices T and S withcolumns t 1 ;:::;t f androws s 1 ;:::;s f ,respectively.Thenwecanwrite  P fi  1 t i s i asthesumof f nonnegativematricesofrankone. Callingthesetofallnon-zeropositionsofamatrixits support ,we thusndthatthenonnegativefactorization  TS providesaway tocoverthesupportof by f rectangles ,i.e.,setsoftheform I J ,where I and J aresubsetsoftherow-andcolumn-indices of ,respectively.Hence,duetoTheorem2,theminimumnumberofrectanglesbywhichonecancoverthesupportof yields alowerbound(the rectanglecoveringbound )ontheextensioncomplexityof P (Yannakakis[ 34 ]).Actually,therectanglecoveringbound dominatesthebounddiscussedinthepreviousparagraph[ 17 ].As Yannakakis[ 34 ]observedfurthermore,thelogarithmoftherectanglecoveringboundofapolytope P isequaltothe nondeterministic communicationcomplexity ofthepredicateonthepairs v;f ofvertices v andfacets f of P thatistrueifandonlyif v 62 f Onecanequivalentlydescribetherectanglecoveringbound as theminimumnumberofcompletebipartitesubgraphsneededt o coverthe vertex-facet-non-incidencegraph ofthepolytope P .A foolingset isasubset F oftheedgesofthisgraphsuchthatnotwoof theedgesin F arecontainedinacompletebipartitesubgraph.Thus everyfoolingset F provesthattherectanglecoveringbound,and hence,theextensioncomplexityof P ,isatleast j F j .Forinstance, forthe n -dimensionalcubeitisnottoodifculttocomeupwitha foolingsetofsize 2 n ,provingthatforacubeonecannotdobetter byallowingextendedformulationsfortherepresentation. Formore detailsonboundsofthistypewereferto[ 17 ]. Unfortunately,allinallthecurrentlyknowntechniquesfo rderivinglowerboundsonextensioncomplexitiesareratherlimit edand yieldmostlyquiteunsatisfyingbounds. 4.2TheRoleofSymmetry Asking,forinstance,abouttheextensioncomplexityofthe matchingpolytope P match n denedinthebeginning,onendsthatnot muchisknown.Itmightbeanythingbetweenquadraticandexp onentialin n .However,inthemainpartofhisstrikingpaper[ 34 ],Yannakakisestablishedanexponentiallowerboundonthesizes of symmetric extendedformulationsof P match n .Here, symmetric means thattheextensionpolyhedronremainsunchangedwhenrenum beringthenodesofthecompletegraph,ormoreformallythat,fo reach permutation oftheedgesofthecompletegraphthatisinducedby apermutationofitsnodes,thereisapermutation ofthevariables oftheextendedformulationthatmapstheextensionpolyhed ronto itselfsuchthat,foreveryvector y intheextendedspace,applying totheprojectionof y yieldsthesamevectorasprojectingthe vectorobtainedfrom y byapplying .Indeed,manyextendedformulationsaresymmetricinasimilarway,forinstancetheex tended formulationofthepermutahedronbytheBirkhoff-polytope mentionedintheIntroductionaswellastheextendedformulati onfor thespanningtreepolytopediscussedinSection2.1. InordertostateYannakakis'resultmoreprecisely,denote by M ` n thesetofallmatchingsofcardinality ` inthecomplete graphwith n nodes,andby P match` n  conv f M : M 2M ` n g theassociatedpolytope.Inparticular, P matchn= 2 n isthe perfectmatching-polytope (foreven n ). Theorem3 (Yannakakis[ 34 ]) Foreven n ,thesizeofevery symmetric extendedformulationof P matchn= 2 n isatleast  n b n 2 = 4 c  Since P matchb n= 2 c n is(isomorphicto)afaceof P match n ,oneeasilyderivestheabovementionedexponentiallowerboundonthesiz esof symmetricextendedformulationsfor P match n fromTheorem3. AtthecoreofhisbeautifulproofofTheorem3,Yannakakissh ows that,foreven n ,thereisnosymmetricextendedformulationin equationform(i.e.,withequationsandnonnegativitycons traints only)of P matchn= 2 n ofsizeatmost n k with k b n 2 = 4 c .From suchahypotheticalextendedformulation EF 1 ,herstconstructsan extendedformulation EF 2 inequationformonvariables y A forall matchings A with j A j k suchthatthe0/1-vectorvaluedmap s ? ontheverticesof P matchn= 2 n denedby s ? M A  1 ifandonly if A M isa section of EF 2 ,i.e., s ? x mapseveryvertex x toa

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6 OPTIMA85 preimageundertheprojectionof EF 2 thatiscontainedintheextensionpolyhedron.Thenitturnsoutthatanextendedformu lation like EF 2 cannotexist.Infact,foranarbitrarypartitioningofthen ode setintotwoparts V 1 and V 2 with j V 1 j 2 k  1 ,onecanconstruct anonnegativepoint y ? intheafnehulloftheimageof s ? (thus y ? iscontainedintheextensionpolyhedronof EF 2 thatisdened byequationsandnonnegativityconstraintsonly)with y ? f e g  0 for alledges e connecting V 1 and V 2 ,whichimpliesthattheprojection ofthepoint y ? violatestheinequality xV 1  1 thatisvalid for P matchn= 2 n (since j V 1 j 2 k  1 isodd).Thecrucialingredientfor constructing EF 2 from EF 1 isatheoremofBocherts'[ 8 ]statingthat everysubgroup G ofpermutationsof m elementsthatisprimitive with j G j >m = b m  1 = 2 c containsallevenpermutations.Yannakakisconstructsasection s for EF 1 forthathecanshowby exploitingBochert'stheoremthatthereisanonnegativem atrix C with sM  C s ? M forall M 2M n= 2 n ,whichmakesit ratherstraightforwardtoconstruct EF 2 from EF 1 Withrespecttothefactthathisproofyieldsanexponential lower boundonlyfor symmetric extendedformulations,Yannakakis[ 34 ]remarkedwedonotthinkthatasymmetryhelpsmuchinconstru ctingsmallextendedformulationsofthe(perfect)matchingp olytopes andstatedasanopenproblemtoprovethatthematching(... ) polytopescannotbeexpressedbypolynomialsizeLP'switho utthe symmetryassumption.Asindicatedabove,todaywestilldo not knowwhetherthisispossible.However,atleastitturnedou trecentlythatrequiringsymmetrycanmakeabigdifferencefor the smallestpossiblesizeofanextendedformulation.Theorem4 ([ 22 ]) Allsymmetricextendedformulationsof P matchb log n c n havesizeatleast n  log n ,whiletherearepolynomialsizenon-symmetric extendedformulationsfor P matchb log n c n (i.e.,theextensioncomplexity of P matchb log n c n isboundedfromabovebyapolynomialin n ). Thus,atleastwhenconsideringmatchingsofsize b log n c insteadof perfect(orarbitrary)matchings,asymmetryindeedhelpsm uch. Whiletheproofofthelowerboundonthesizesofsymmetric extendedformulationsstatedinTheorem4isamodicationo fYannakakis'proofindicatedabove,theconstructionofthepol ynomial sizenon-symmetricextendedformulationof P matchb log n c n relieson theprincipleofdisjunctiveprogramming(seeSection2.2) .Foran arbitrarycoloring ofthe n nodesofthecompletegraphwith 2 k colors,wecallamatching M (with j M j k ) -colorful if,ineach ofthe 2 k colorclasses,thereisexactlyonenodethatisanendnodeofoneoftheedgesfrom M .Letusdenoteby P theconvex hullofthecharacteristicvectorsof -colorfulmatchings.Thecrucial observationisthat P canbedescribedby O  2 k  n 2  inequalities (asopposedto  2 n  inequalitiesneededtodescribethepolytope associatedwithallmatchings,seetheIntroduction).Onth eother hand,accordingtoatheoremduetoAlon,Yuster,andZwick[ 2 ], thereisafamilyof q suchcolorings 1 ;:::; q with q  2 O k log n suchthat,forevery 2 k -elementsubset W ofthe n nodes,inatleast oneofthecoloringsthenodesfrom W receivepairwisedifferent colors.Thuswehave P matchk n  conv P 1 [[ P q  ,andhence (asdescribedinSection2.2)weobtainanextendedformulat ion of P matchk n ofsize 2 O k n 2 log n ,which,for k b log n c ,yieldsthe upperboundinTheorem4. Yannakakis[ 34 ]moreoverdeducedfromTheorem3thatthere arenopolynomialsizesymmetricextendedformulationsfor the travelingsalesmanpolytope(theconvexhullofthecharact eristic vectorsofallcyclesoflengths n inthecompletegraphwith n nodes).SimilarlytoTheorem4,onecanalsoprovethatthere are nopolynomialsizesymmetricextendedformulationsforthe polytopesassociatedwithcyclesoflength b log n c ,whilethesepolytopes neverthelesshavepolynomiallyboundedextensioncomplex ity[ 22 ]. Pashkovich[ 27 ]furtherextendedYannakakis'techniquesinorder toprovethateverysymmetricextendedformulationofthepe rmutahedron P perm n hassizeatleast n 2  ,showingthattheBirkhoffpolytopeessentiallyprovidesanoptimal symmetric extensionforthe permutahedron. 5Conclusions Manypolytopesassociatedwithcombinatorialoptimizatio nproblemscanberepresentedinsmall,simple,andnicewaysaspro jectionsofhigherdimensionalpolyhedra.Moreover,thoughwe have nottouchedthistopichere,sometimessuchextendedformul ations arealsoveryhelpfulinderivingdescriptionsintheorigin alspaces. Whatwecurrentlylackareontheonehandmoretechniquestoconstructextendedformulationsandontheotherhandagood understandingofthefundamentallimitsofsuchrepresentati ons.For instance,doeseverypolynomiallysolvablecombinatorial optimizationproblemadmitanextendedformulationofpolynomialsi ze?We evendonotknowthisforthematchingproblem.Howaboutthestablesetprobleminperfectgraphs?Thebestupperboundon the extensioncomplexityofthesepolytopesforgraphswith n nodes stillis n O  log n (Yannakakis[ 34 ]). Progressonsuchquestionswilleventuallyshedmorelighto nto theprinciplepossiblitiestoexpresscombinatorialprobl emsby meansoflinearconstraints.Moreover,thesearchforexten dedformulationsyieldsnewmodellingideassomeofwhichmayprove to beusefulalsoinpracticalcontexts.Inanycase,workonext ended formulationscanleadintofascinatingmathematics.Acknowldgements. WearegratefultoSamBurer,Kanstantsin Pashkovich,BrittaPeis,LaurenceWolsey,andMihalisYann akakisfor commentsonadraftofthisarticleandtoMatthiasWalterfor producingFigure1.VolkerKaibel,InstitutfrMathematischeOptimierung,Fa kulttfrMathematik,Otto-von-GuerickeUniversittMagdeburg,Univ ersittsplatz2, 39106Magdeburg,Germany. kaibel@ovgu.de References [1]M.Ajtai,J.Komls,andE.Szemerdi.Sortingin c log n parallelsteps. Combinatorica ,3(1):119,1983. [2]NogaAlon,RaphaelYuster,andUriZwick.Color-coding. J.Assoc.Comput. Mach. ,42(4):844856,1995. [3]EgonBalas.Disjunctiveprogrammingandahierarchyofr elaxationsfordiscreteoptimizationproblems. SIAMJ.AlgebraicDiscreteMethods ,6(3):466 486,1985. [4]EgonBalasandWilliamPulleyblank.Theperfectlymatch ablesubgraph polytopeofabipartitegraph.In Proceedingsofthesymposiumonthematchingproblem:theory,algorithms,andapplications(Gaithe rsburg,Md.,1981) volume13,pages495516,1983. [5]FranciscoBarahona.Oncutsandmatchingsinplanargrap hs. Math.Programming ,60(1,Ser.A):5368,1993. [6]AharonBen-TalandArkadiNemirovski.Onpolyhedralapp roximationsof thesecond-ordercone. Math.Oper.Res. ,26(2):193205,2001. [7]GarrettBirkhoff.Threeobservationsonlinearalgebra Univ.Nac.Tucumn. RevistaA. ,5:147151,1946. [8]AlfredBochert.UeberdieZahlderverschiedenenWerthe ,dieeineFunctiongegebenerBuchstabendurchVertauschungderselbener langenkann. Math.Ann. ,33(4):584590,1889. [9]KevinKingHinCheung. Subtoureliminationpolytopesandgraphsofinscribabletype .ProQuestLLC,AnnArbor,MI,2003.Thesis(Ph.D.)Univers ity ofWaterloo(Canada).

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April2011 9 [9]A.FrangioniandB.Gendron.0-1reformulationsofthemu lticommoditycapacitatednetworkdeignproblem. DiscreteAppliedMathematics 157:12291241,2009. [10]R.Fukosawa,H.Longo,J.Lysgaard,M.Reis,E.Uchoa,an dR.F.Werneck. Robustbranch-and-cut-and-priceforthecapacitatedvehi cleroutingproblem. MathematicalProgramming ,106:491511,2006. [11]L.Gouveia.A2n-constraintformulationforthecapaci tatedminimalspanningtreeproblem. OperationsResearch ,43:130141,1995. [12]O.GnlkandY.Pochet.Mixingmixedintegerinequalit ies. Mathematical Programming ,90:429457,2001. [13]J.KrarupandO.Bilde.Plantlocation,setcoveringand economiclotsizes: An Omn algorithmforstructuredproblems.InL.Collatzetal.,edi tor, OptimierungbeiGraphentheoretischenundGanzzahligenPr obleme ,pages 155180.BirkhauserVerlag,Basel,1977. [14]T.L.Magnanti,P.Mirchandani,andR.Vachani.Theconv exhulloftwocore capacitatednetworkdesignproblems. MathematicalProgramming ,60:233 250,1993. [15]R.MeloandL.A.Wolsey.Mipformulationsandheuristic sfortwo-level production/transportationproblems.Technicalreport,C ORE,Universit catholiquedeLouvain,2011. [16]A.Pessoa,E.Uchoa,M.P.deAragao,andR.Rodrigues.Ex actalgorithm overanarc-time-indexedformulationforparallelmachine schedulingproblems. MathematicalProgrammingComputation ,2:259290,2010. [17]Y.PochetandL.A.Wolsey.Lot-sizingwithconstantbat ches:Formulationandvalidinequalities. MathematicsofOperationsResearch ,18:767785, 1993. [18]Y.PochetandL.A.Wolsey. ProductionPlanningbyMixedIntegerProgramming Springer,2006. [19]A.A.B.Pritsker,L.J.Watters,andP.J.Wolfe.Multipr ojectschedulingwith limitedresources:azero-oneprogrammingapproach. ManagementScience 16:93108,1969. [20]E.Uchoa,R.Fukasawa,J.Lysgaard,A.Pessoa,M.P.Arag ao,andD.Andrade. Robustbranch-and-cut-and-priceforthecapacitatedmini mumspanning treeproblemoveranextendedformulation. MathematicalProgramming 112:563591,2004. [21]M.VanVyveandL.A.Wolsey.Approximateextendedformu lations. MathematicalProgrammingB ,105:501522,2006. [22]F.VanderbeckandL.A.Wolsey.Reformulationanddecom positionofintegerprograms.InM.Jngeretal.,editor, 50YearsofIntegerProgramming 1958-2008 ,pages431502.Springer,2010. [23]R.T.Wong.Integerprogrammingformulationsofthetra velingsalesman problem. Proceedingsof1980IEEEInternationalConferenceonCircu itsand Computers ,xx:149152,1980. MihalisYannakakisOnExtendedLPFormulations Inconnectionwithhispaper,VolkerKaibelaskedmetogives ome backgroundonmypaperExpressingCombinatorialOptimiza tion ProblemsbyLinearPrograms"[ 3 ]regardingthemotivationand thoughtsthatguidedthatwork.Thatresearchwascarriedou tin 1987andwaspresentedrstattheSTOC'88conference.Itwas motivatedontheonehandbyaclaimedproofofP=NPthatappeare d atthattimeandattractedalotofattentioninthecommunity ,and ontheotherhandbythedevelopmentsintheprecedingyearsi n LinearProgrammingandthepolyhedralapproachtocombinat orial optimization. In198687E.R.Swartcirculatedapaperthatclaimedtosolv ethe TravelingSalesmanProblemusingLinearProgramming[ 2 ].Inparticular,thepaperconstructedalargeLPwithmanyextravariab les( n 8 intheoriginalversion, n 10 inarevision),whichpurportedlyprovidedanextendedformulationoftheTSPpolytope.Giventhe size andcomplexityoftheLP,itwasratherhardtodeterminewhat was exactlytheeffectofthesevariablesandconstraintsandwh etherthe LPworkedornot.Itwasclearthatsomemethodologywasneede d tounderstandwhatispossibletoachievewithextravariabl esand whetherthisapproachcouldpossiblyworkinprinciple. Theapproachitselfisactuallyareasonableonetotryifone believesthatP=NP.First,LinearProgrammingisaP-complete problem,soitisinsomesenseahardest,universalprobleminP.S econd,weknowthattheintroductionofextravariables(which are thenexistentiallyquantied,i.e.projectedout)isapowe rfultoolin variousdomainsthatcanincreasedrasticallytheexpressi vepower ofamodel,turninganexponentialobjectintoapolynomialo ne. Thereareforexamplesimplepolytopeswithanexponentialn umberoffacets,thatcanbeexpressedsuccinctlywithextrava riables. Inlogic,everyBooleanpredicatecanberepresentedbyaneq uivalentBooleanformula(inconjunctivenormalform),butthe formularequiresoftenexponentialsize.However,introducin gextra variableswecanexpressanyNPpredicatebyapolynomial-si zeformula;thisisessentiallyCook'stheoremshowingthatSatis abilityis NP-complete. AnothermotivationcamefromthediscoveryofKhachian'sel lipsoidalgorithmanditsapplications,andKarmakar'salg orithm. Grtchel,LovaszandSchrijverhadshownthattheellispoid algorithmcouldbeusedtosolveinpolynomialtimevariousprobl ems (suchasthecliqueandindependentsetproblemonperfectgr aphs) eventhoughtheirstandardLPformulationhasanexponentia lnumberofconstraints(providedthereisagoodseparationalgo rithmfor theconstraints).Inviewoftheimpracticalityoftheellip soidalgorithm,itwouldbedesirabletouseinsteadKarmakar'salgor ithm(or Simplex)fortheseproblems;howeverthiswouldrequireapo lynomialsizeLPdescription,whichraisesthequestionwhether onecan constructsuchaformulationusingany(small)setofextrav ariables andconstraints.ThesamequestionisrelevantalsoforNP-h ard problems,suchastheTSP,withrespecttousefulclassesoft heir facets;presumably(ifP  NP)wecannotconstructexactextended formulationsinpolynomialtime,butperhapswecanexpress succinctlyimportantclassesoffacetsthatprovideusefulcut tingplanes. WiththisbackgroundItriedtotakeasystematiclooktoget someunderstandingofwhatcanandcannotbeachievedwithco mpactextendedLPformulations.Thepaperresolvedpartiall ysome questionsandleftmanymoreopen.Forgeneralpolytopesapl easantsurprisewasthatthecomplexityofthesmallestextende dLP formulationcanbecharacterizedbyaconcreteparameter,t henonnegativerankoftheslackmatrix.Theproblemisthatitisof ten notthateasytocomputethenonnegativerank.Thepaperpoin ted outaconnectiontocommunicationcomplexity,whichcanbeu sed sometimestoobtainlowerorupperbounds,andappliedittot he independent(stable)setpolytopeforsomeclassesofgraph s.Ingeneralhowever,westillneedtodevelopgoodmethodstoestima teor boundthenonnegativerank. WithrespecttothequestionofexpressingtheTSPpolytopeb y asmallLP,Icouldnotshowthatthisisimpossible,butcould ruleit outatleastfortheclassofsymmetricLPs(whichincludesth eproposedLPinSwart'sattemptedproof):anysuchformulationm ust haveexponentialsize.Itwaseasiertoprovethisresultrs tforthe (perfect)matchingpolytopeandthentransferitbyreducti ontothe TSPpolytope.(Theresultholdsforasomewhatlargerclasso fLPs butitwassimplertostateitforsymmetricLPs.)Insubseque ntyears therehavebeenseveralmoresimilarattemptstoproveP=NPu sing anextendedLPformulationfortheTSP;theyaregenerallysy mmetricoralmostso(sometimesforexampleaparticularnodeiss ingled outasthestartingnodeofthetour;theexponentiallowerbo und holdsevenifaconstantfractionofnodesissingledout.)Sy mmetry seemstoberathernaturalintheconstructionofexactLPfor mulations,butasshownrecentlybyKaibel,PashkovichandThe is[ 1 ], nonsymmetrycansaveasuperpolynomialamountinsomecases Itremainsanintriguingopenquestionwhetherthematching polytopecanbeexpressedbyapolynomial-size(unrestricted)L P.Forthe

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OPTIMA

Philippe L. Toint
MOS Chair's Column
March 30, 201 I. The Mathematical Optimization Society is alive
and active, and the best proof of this statement is the number of
innovations that have taken place within the Society since I wrote
my last column in Optima. Let me review these innovations briefly.
The first innovation, especially from the researcher point of view,
is the new internet support which has been introduced for our main
scientific journal, Mathematical Programming (Series A). This change
(described in another column in this issue of Optima) aims at pro-
viding better service to the optimization research community, and, I
am certain, will be beneficial for this community.
The second important event I wish to bring to your attention is
the creation of "Named Lectureships" at the Mathematical Program-
ming International Symposium. These lectureships are intended to
honour prominent researchers in optimization and related fields by
selecting a scientist to give a special lecture during the Symposium
(and possibly attributing a cash prize associated with the lecture-
ship). Fittingly, the first such named lectureship, which has been ac-
cepted by the Council of the Society, is the "Paul Y. Tseng Memorial
Lectureship in Continuous Optimization". It will be presented for
the first time at the Twenty First International Symposium of Math-
ematical Programming (ISMP) in 2012, and triennally at each ISMP
thereafter. This lectureship was established on the initiative of fam-
ily and friends of Professor Tseng, with financial contributions to
the associated endowment also coming from universities and com-
panies in the Asia Pacific region. The purposes of the lectureship are
to commemorate the outstanding contributions of Professor Tseng
in continuous optimization and to promote the research and appli-
cations of continuous optimization in the Asia Pacific region. The
Council of the Society has not excluded that other named lecture-
ships could be established in the future, and has modified the bylaws
of the Society (available on the MOS website) to establish suitable
rules and conditions.
Needless to say, I am extremely pleased and proud of this new
valuable development, and I am truly looking forward to the first
Paul Tseng Lecture in Berlin.
A third potentially important decision has been taken by the MOS
Council regarding the collection of the Society's membership fees.
It is a very unfortunate fact that so far the MOS membership falls
significantly once the memberships granted during the International
Symposium expire. As it turns out, many members forget to re-
new their membership and this puts the whole Society in a difficult
situation, because this happens in the period during which the next
Symposium is being organized. During this period a healthy member-
ship is needed to give the organizers a stronger negotiating position
with local facilitators, such as hotels or universities. The council has
therefore decided to propose financially attractive multi-year mem-
bership packages during the registration process of the Symposium.
The first such offer will be made available at the registration for the
Berlin 2012 Symposium.

This last topic provides the necessary transition for my (admit-
tedly repeated) urgent call to all: Please consider being a MOS member,
or to renew your membership for 201 I if you have not yet done so.
We need a strong society to represent our research community and
help the organizers of the fantastic forthcoming Berlin Symposium.

Note from the Editors

Dear MOS Members,
We are pleased to take over the editorship of the Optima newslet-
ter from the extremely capable team of editor Andrea Lodi and
co-editors Alberto Caprara and Katya Scheinberg (who has been
promoted to editor). We thank them for their tremendous service!
We want to briefly introduce the new team, which is as ge-
ographically and scientifically diverse as ever. Katya Scheinberg
(katyas@lehigh.edu) works in various areas of continuous optimiza-
tion with special interests in derivative free optimization and appli-
cations of statistical learning. Sam Burer (samuel-burer@uiowa.edu)
conducts research in convex optimization and strives to write fast,
accurate optimization code. Volker Kaibel's (kaibel@ovgu.de) main
scientific interest is in discrete optimization with special emphasis
on polyhedral aspects.
As the editorial board, our hope is to continue Optima as a timely
newsletter with high quality contributions. We will retain the format
established by the previous board with one main scientific contri-
bution per issue and a discussion column on a related topic. We
hope that the readers will continue finding this format stimulating.
Please let us know if you have a contribution yourself or if there is a
particular topic you would like to see covered in Optima.
In this issue we present an article by our own new co-editor,
Volker Kaibel, and two discussion pieces all on the topic of extended
formulations.
Katya Scheinberg, Editor
Sam Burer, Co-Editor
Volker Kaibel, Co-Editor

Contents of Issue 85 / April 201 I
I Philippe L. Toint, MOS Chair's Column
I Note from the Editors
2 Volker Kaibel, Extended Formulations in
Combinatorial Optimization
7 Laurence A. Wolsey, Using Extended Formulations in Practice
9 Mihalis Yannakakis, On Extended LP Formulations
10 Mathematical Programming, Series A, is going online
10 Call for nomination for the 2012 George B. Dantzig Prize
10 Conference Announcements
II Imprint

OPTIMA 85

Volker Kaibel

Extended Formulations in

Combinatorial Optimization

I Introduction
Linear Programming based methods and polyhedral theory form the
backbone of large parts of Combinatorial Optimization. The basic
paradigm here is to identify the feasible solutions to a given prob-
lem with some vectors in such a way that the optimization problem
becomes the problem of optimizing a linear function over the fi-
nite set X of these vectors. The optimal value of a linear function
over X is equal to its optimal value over the convex hull conv(X) =
{xcx AxX : Yxex Ax = 1, A > } of X. According to the Weyl-
Minkowski Theorem [33, 25], every polytope (i.e., the convex hull of
a finite set of vectors) can be written as the set of solutions to a
system of linear equations and inequalities. Thus one ends up with a
linear programming problem.
As for the maybe most classical example, let us consider the set
M(n) of all matching in the complete graph Kn = (Vn,En) on n
nodes (where a matching is a subset of edges no two of which share
a common end-node). Identifying every matching M c En with its
characteristic vector x(M) e {0, 1}En (where x(M)e 1 if and only
if e cM), we obtain the matching polytope pmatch(n) = conv{x(M) :
M e M(n)} In one of his seminal papers, Edmonds [13] proved
that pmatch(n) equals the set of all x e E"n that satisfy the inequal-
ities x(6(v)) < 1 for all v e V, and x(En(S)) < [ISI/2] for all
subsets S c Vn of odd cardinality 3 < |SI < n (where 6(v) is the
set of all edges incident to v, En(S) is the set of all edges with both
end-nodes in S, and x(F) = ZeeFxe). No inequality in this system,
whose size is exponential in n, is redundant.
The situation is quite similar for the permutahedron PPerm(n),
i.e., the convex hull of all vectors that arise from permuting the
components of (1, 2...,n). Rado [29] proved that PPerm(n) is
described by the equation x([n]) = n(n + 1)/2 and the in-
equalities x(S) > ISl(ISl + 1)/2 for all 0 1 S s [n] (with
[n] {= ..., n}), none of the 2n 2 inequalities being redun-
dant. However if for each permutation a : [n] [n] we con-
sider the corresponding permutation matrix y e {0, 1}" n (sat-
isfying yij = 1 if and only if a(i) = j) rather than the vec-
tor (ar(1),...,ar(n)), we obtain a much smaller description of
the resulting polytope, since, according to Birkhoff [7] and von
Neumann [32], the convex hull pbirk(n) (the Birkhoff-Polytope) of
all n x n-permutation matrices is equal to the set of all doubly-
stochastic n x n-matrices (i.e., nonnegative n x n-matrices all of
whose row- and column sums are equal to one). It is easy to
see that the permutahedron pPerm (n) is a linear projection of
the Birkhoff-polytope pbirk(n) via the map defined by p(y)i =
Sj1 Jyij. Since, for every linear objective function vector ce IRn,
we have max{(c,x) : x PPerm(n)} max{!i I i jCiYij :
y e pbik(n)}, one can use pbirk(n) (that can be described by n2
nonnegativity inequalities) instead of pPerm(n) (whose description
requires 2n -2 inequalities) with respect to linear programming re-
lated issues.
In general, an extension of a polytope P c R" is a polyhe-
dron Q c Rd (i.e., an intersection of finitely many affine hyper-
planes and halfspaces) together with a linear projection p : R RVn
satisfying P = p(Q). Any description of Q by linear equations and
linear inequalities then (together with p) is an extended formulation
of P. The size of the extended formulation is the number of in-
equalities in the description. Note that we neither account for the

number of equations (we can get rid of them by eliminating vari-
ables) nor for the number of variables (we can ensure that there
are not more variables than inequalities by projecting Q to the or-
thogonal complement of its lineality space, where the latter is the
space of all directions of lines contained in Q). If T e nx1d is
the matrix with p(y) = Ty, then, for every ce Rn, we have
max{(c,x) : x P max(Ttc,y) : y Q}.
In the example described above, pbirk(n) thus provides an ex-
tended formulation of pPerm (n) of size n2. It is not known whether
one can do something similar for the matching polytopes match (n)
(we will be back to this question in Section 4.2). However there
are many other examples of nice and small extended formulations
for polytopes associated with combinatorial optimization problems.
light on the geometric, combinatorial and algebraic background of
this concept that recently has received increased attention. The pre-
sentation is not meant to be a survey (for this purpose, we refer to
Vanderbeck and Wolsey [31] as well as to Cornu6jols, Conforti,
and Zambelli [I I]) but rather an appetizer for investigating alterna-
tive possibilities to express combinatorial optimization problems by
means of linear programs.
While we will not be concerned with practical aspects here, ex-
tended formulations have also proven to be useful in computations.
You will find more on this in Laurence Wolsey's discussion column
below. Fundamental work with respect to understanding the con-
cept of extended formulations and its limits has been done by Mihalis
Yannakakis in his 1991-paper Expressing Combinatorial Optimization
Problems by Linear Programs [34] (see Sect. 3.3 and 4). We are very
happy that he shares with us some of his thoughts on the subject in
another discussion column.

2 Some Examples
2. 1 Spanning Trees
The spanning tree polytope Pspt(n) associated with the complete
graph Kn = (Vn, En) on n nodes is the convex hull of all character-
istic vectors of spanning trees, i.e., of all subsets of edges that form
connected and cycle-free subgraphs. In another seminal paper, Ed-
monds [ 14] proved that pspt(n) is the set of all x e RnE that satisfy
the equation x(En) n 1- and the inequalities x(En(S)) < IS -I 1
for all S c Vn with 2 < |SI < n. Again, none of the exponentially
many inequalities is redundant.
However, by introducing additional variables z,,,,,,u for all or-
dered triples (v, w, u) of pairwise different nodes meant to encode
whether the edge {v,w} is contained in the tree and u is in the
component of w when removing {v,w} from the tree, it turns
out that the system consisting of the equations x{,,,, zv,w,u -
zw,,Vu, = 0 and x{v,w} + Yuc[n}\Jv,,w} Zv,,,w = 1 (for all pairwise
different v,w,u e Vn) along with the nonnegativity constraints
and the equation x(En) n 1 provides an extended formula-
tion of Pspt(n) of size 0(n3) (with orthogonal projection to the
space of x-variables). This formulation is due to Martin [23] (see
also [34, I I]). You will find an alternative one in Laurence Wolsey's
discussion column below.

2.2 Disjunctive Programming
If Pi c Vn is a polytope for each i e [q], then clearly P
conv(Pi u ... u Pq) is a polytope as well, but, in general, it is diffi-
cult to derive a description by linear equations and inequalities in Rn
from such descriptions of the polytopes Pi. However constructing
an extended formulation for P in this situation is very simple. In-
deed suppose that each Pi is described by a system Aix < bi of fi
linear inequalities (where, in order to simplify notation, we assume

April 201 I1

that equations are written, e.g., as pairs of inequalities). Then the
system Aizz < Aibi for all i e [q], _i_ Ai = 1, A > 0 with vari-
ables zi e R" for all i e [q] and A Ge R is an extended formulation
for P of size fi + - + fq + q, where the projection is given by
(z1 ..., zI, A) z1 + . + zq. This has been proved first by Balas
(see, e.g., [3]), even for polyhedra that are not necessarily polytopes
(where in this general case P needs to be defined as the topological
closure of the convex hull of the union).

2.3 Dynamic Programming
When a combinatorial optimization problem can be solved by a dy-
namic programming algorithm, one often can derive an extended for-
mulation for the associated polytope whose size is roughly bounded
by the running time of the algorithm.
A simple example is the 0/1-Knapsack problem, where we are
given a nonnegative integral weight vector w e Nn, a weight
bound W e N, and a profit vector c e R", and the task is to
solve max{(c,x) : x cF(w,W)} with F(w,W) = {x {0,1} :
(w,x) < W A classical dynamic programming algorithm works by
setting up an acyclic directed graph with nodes s = (0, 0), t, and
(i, w) for all i e [n], w e {0, 1,... W} and arcs from (i, w) to
(i', w') if and only if i < i' and w' = w + wi, where such an arc
would be assigned length ci,, as well as arcs from all nodes to t (of
length zero). Then solving the 0/1-Knapsack problem is equivalent
to finding a longest s-t-path in this acyclic directed network, which
can be carried out in linear time in the number a of arcs.
The polyhedron Q c RI of all s-t-flows of value one in that
network equals the convex hull of all characteristic vectors of s-t-
paths (due to the total unimodularity of the node-arc incidence ma-
trix), thus it is easily seen to be mapped to the associated Knapsack-
polytope pknap(, W) conv(F(w, W)) via the projection given by
y x, where xi is the sum of all components of y indexed by
arcs pointing to nodes of type (i, *). As Q is described by nonneg-
ativity constraints, the flow-conservation equations on the nodes
different from s and t and the equation ensuring an outflow of value
one from s, these constraints provide an extended formulation for
pknap(W, W) of size a.
However quite often dynamic programming algorithms can only
be formulated as longest-paths problems in acyclic directed hy-
pergraphs with hyperarcs of the type (S, v) (with a subset S of
nodes) whose usage in the path represents the fact that the opti-
mal solution to the partial problem represented by node v has been
constructed from the optimal solutions to the partial problems rep-
resented by the set S. Martin, Rardin, and Campbell [24] showed
that, under the condition that one can assign appropriate reference
sets to the nodes, also in this more general situation nonnegativity
constraints and flow-equations suffice to describe the convex hull of
the characteristic vectors of the hyperpaths. This generalization al-
lows one to derive polynomial size extended formulations for many
of the combinatorial optimization problems that can be solved in
polynomial time by dynamic programming algorithms.

2.4 Others
A common generalization of the techniques to construct extended
formulations by means of disjunctive programming or dynamic pro-
gramming is provided by branched polyhedral systems (BPS) [20]. In
this framework, one starts from an acyclic directed graph that has as-
sociated with each of its non-sink nodes v a polyhedron in the space
indexed by the out-neighbors of v. From these building blocks, one
constructs a polyhedron in the space indexed by all nodes. Under
certain conditions one can derive an extended formulation for the
constructed polyhedron from extended formulations of the polyhe-
dra associated with the nodes.

Some very nice extended formulations have recently been given
by Faenza, Oriolo, and Stauffer [ 16] for stable set polytopes of claw-
free graphs. Here the crucial step is to glue together descriptions
of stable set polytopes of certain building block graphs by means
of strip compositions. One of their constructions can be obtained by
applying the BPS-framework, though apparently the most interesting
one they have cannot.
An asymptotically smallest possible extended formulation of size
O(nlog n) for the permutahedron pperm (n) has been found by Goe-
mans [18]. His construction relies on the existence of sorting net-
works of size r = O(nlogn) (Ajtai, Koml6s, and Szemer6di [I]),
i.e., sequences (iL,Ji),..., (ir,j) for which the algorithm that in
each step s swaps elements i, and i, if and only if .1 > .i,
sorts every sequence (al,...,an) e [R into non-decreasing order.
The construction principle of Goemans has been generalized to the
framework of reflection relations [21], which, for instance, can be
used to obtain small extended formulations for all G-permutahedra
of finite reflection groups G (see, e.g., Humphreys [19]), including ex-
tended formulations of size 0(logm) of regular m-gons, previously
constructed by Ben-Tal and Nemirovski [6]. Another application of
reflection relations yields extended formulations of size O(n log n)
for Huffman-polytopes, i.e., the convex hulls of the leaves-to-root-
distances vectors in rooted binary trees with n labelled leaves. Note
that linear descriptions of these polytopes in the original spaces are
very large, rather complicated, and unknown (see Nguyen, Nguyen,
and Maurras [26]).
The list of combinatorial problems for which small (and nice)
extended formulations have been found comprises many others,
among them perfect matching polytopes of planar graphs (Bara-
hona [5]), perfectly matchable subgraph polytopes of bipartite
graphs (Balas and Pulleyblank [4]), stable-set polytopes of dis-
tance claw-free graphs (Pulleyblank and Shepherd [28]), packing
and partitioning orbitopes [I5], subtour-elimination polytopes (Yan-
nakakis [34] and, for planar graphs, Rivin [30], Cheung [9]), and cer-
tain mixed-integer programs (see, e.g., Conforti, di Summa, Eisen-
brand, and Wolsey [ 12]).

3 Combinatorial, Geometric, and Algebraic
Background
3.1 Face Lattices
Any intersection of a polyhedron P with the boundary hyperplane
of some affine halfspace containing P is called a face of P. The empty
set and P itself are considered to be (non-proper) faces of P as
well. The proper faces of a three-dimensional polytope thus are its
vertices, edges, and the polygons that make up the boundary of P.
Partially ordered by inclusion, the faces of a polyhedron P form a
lattice (P), the face lattice of P. The proper faces that are maximal
with respect to inclusion are the facets of P. Equivalently, the facets
of P are those faces whose dimension is one less than the dimen-
sion of P. An irredundant linear description of P has exactly one
inequality for each facet of P.
If Q c YRd is an extension of the polytope P c R with a lin-
ear projection p : R Rn, then mapping each face of P to its
preimage in Q under p defines an embedding of (P) into (Q).
Figure I illustrates this embedding for the trivial extension Q =
{y e : v x+ x y XYx 1} of P = conv(X) via p(y) XY x yxX
for X = {1, .... ,4, -4} (thus P is the cross-polytope in IR4
with 16 facets and Q is the standard-simplex in R8 with 8 facets). As
this figure suggests, constructing a small extended formulation for a
polytope P means to hide the facets of P in the fat middle part of
the face lattice of an extension with few facets.

OPTIMA 85

Figure I. Embedding of the face lattice of the 4-dimensional cross-polytope into the face lattice of the 7-dimensional simplex

3.2 Slack Representations
Let P {x e A : Ax < b} c IRn be a polytope with affine hull A -
aff(P), A e IRen, and b e R'm. The affine map pq : A IR with
9 (x) b Ax (the slack map of P w.r.t. Ax < b) is injective. We
denote its inverse (the inverse slack map) on its image, the affine sub-
space A= q(A) c Rm, by ( : A A. The polytope P -An RA m,
the slack-representation of P w.r.t. Ax < b, is isomorphic to P with
qp(P) -P and (p(P) -P.
If Z c IRp is a finite set of nonnegative vectors whose convex conic
hull ccone(Z) {zz AzZ : A > 0} c IRm contains P RAn IR,
then we have P Anccone(Z), and thus, the system Yzez Azz e A
and Az > 0 (for all z e Z) provides an extended formulation of P
of size I|Z| via the projection A (p(yzez AzZ). Let us call such an

extension a slack extension and the set Z a slack generating set of P
(both w.r.t. Ax < b).
Now suppose conversely that we have any extended formulation
of P of size q defining an extension Q that is pointed (i.e., the poly-
hedron Q does not contain a line). As for polytopes above (which
in particular are pointed polyhedra), we can consider a slack repre-
sentation Q c Rq of Q and the corresponding inverse slack map (.
Then we have qp(p(Q(Q))) P, where p is the projection map of
the extension. If the system Ax < b is binding for P, i.e., each of
its inequalities is satisfied at equation by some point from P, then
one can show (by using strong LP-duality) that there is a nonnega-
tive matrix T e c with qp(p(f(2))) = T2 for all 2 e Q, thus
P TQ. Hence the columns of T form a slack generating set of P

April 201 I1

(w.r.t. Ax < b), yielding a slack extension of size q. As every non-
pointed extension of a polytope can be turned into a pointed one
of the same size by projection to the orthogonal complement of
the lineality space, we obtain the following result, where the exten-
sion complexity of a polytope P is the smallest size of any extended
formulation of P.

Theorem I ([17]). The extension complexity of a polytope P is equal
to the minimum size of all slack extensions of P.

As every slack extension of a polytope is bounded (and since all
bounded polyhedra are polytopes), Theorem I implies that the ex-
tension complexity of a polytope is attained by an extension that
is a polytope itself. Furthermore, in Theorem I one may take the
minimum over the slack extensions w.r.t. any fixed binding system of
inequalities describing P. In particular, all these minima concide.

3.3 Nonnegative Rank
Now letP conv(X) x e aff(P) : Ax < b} c Rn be a polytope
with some finite set X c Rn and A e R"n, b e R'. The slack ma-
trix of P w.r.t. X and Ax < b is e F. ,',1 with 4i,, b- (Ai,, x).
Thus the slack representation P c Rm of P (w.r.t. Ax < b) is the
convex hull of the columns of 1. Consequently, if the columns of a
nonnegative matrix T ci F form a slack generating set of P,
then there is a nonnegative matrix S e BR x with \$ TS. Con-
versely, for every factorization T'S' of the slack matrix into
nonnegative matrices T' e 1 T f'] and S' e R x, the columns
of T' form a slack generating set for P.
Therefore constructing an extended formulation of size f for P
amounts to finding a factorization of the slack matrix =- TS into
nonnegative matrices T with f columns and S with f rows. In par-
ticular, we have derived the following result that essentially is due to
Yannakakis [34] (see also [ 17]). Here, the nonnegative rank of a ma-
trix is the minumum number f such that the matrix can be written
as a product of two nonnegative matrices, where the first one has f
columns and the second one has f rows.

Theorem 2. The extension complexity of a polytope P is equal to the
nonnegative rank of its slack matrix (w.r.t. any set X and binding system
Ax
Clearly, the nonnegative rank of a matrix is bounded from below
by its usual rank as known from Linear Algebra. There is also quite
some interest in the nonnegative rank of (not necessarily slack) ma-
trices in general (see, e.g., Cohen and Rothblum [10]).

4 Fundamental Limits
4.1 General Lower Bounds
Every extension Q of a polytope P has at least as many faces as P, as
the face lattice of P can be embedded into the face lattice of Q (see
Sect. 3.1). Since each face is the intersection of some facets, one
finds that the extension complexity of a polyhedron with / faces is
at least log /. This observation has first been made by Goemans [ 18]
in order to argue that the extension complexity of the permutahe-
dron pperm (n) is at least O (n log n).
Suppose that TS is a factorization of a slack matrix \$ of
the polytope P into nonnegative matrices T and S with columns
t1,... ,t and rows s1,...,sf, respectively. Then we can write
\$) = tisi as the sum of f nonnegative matrices of rank one.
Calling the set of all non-zero positions of a matrix its support, we
thus find that the nonnegative factorization TS provides a way
to cover the support of \$ by f rectangles, i.e., sets of the form
I x J, where I and J are subsets of the row- and column-indices

of 4, respectively. Hence, due to Theorem 2, the minimum num-
ber of rectangles by which one can cover the support of \$ yields
a lower bound (the rectangle covering bound) on the extension com-
plexity of P (Yannakakis [34]). Actually, the rectangle covering bound
dominates the bound discussed in the previous paragraph [17]. As
Yannakakis [34] observed furthermore, the logarithm of the rectan-
gle covering bound of a polytope P is equal to the nondeterministic
communication complexity of the predicate on the pairs (v, f) of ver-
tices v and facets f of P that is true if and only if v t f.
One can equivalently describe the rectangle covering bound as
the minimum number of complete bipartite subgraphs needed to
cover the vertex-facet-non-incidence graph of the polytope P. A fool-
ing set is a subset F of the edges of this graph such that no two of
the edges in F are contained in a complete bipartite subgraph. Thus
every fooling set F proves that the rectangle covering bound, and
hence, the extension complexity of P, is at least |F|. For instance,
for the n-dimensional cube it is not too difficult to come up with a
fooling set of size 2n, proving that for a cube one cannot do better
by allowing extended formulations for the representation. For more
details on bounds of this type we refer to [ 17].
Unfortunately, all in all the currently known techniques for deriv-
ing lower bounds on extension complexities are rather limited and
yield mostly quite unsatisfying bounds.

4.2 The Role of Symmetry
ing polytope match (n) defined in the beginning, one finds that not
much is known. It might be anything between quadratic and expo-
nential in n. However, in the main part of his striking paper [34], Yan-
nakakis established an exponential lower bound on the sizes of sym-
metric extended formulations of pmatch(n). Here, symmetric means
that the extension polyhedron remains unchanged when renumber-
ing the nodes of the complete graph, or more formally that, for each
permutation Tr of the edges of the complete graph that is induced by
a permutation of its nodes, there is a permutation Kr of the variables
of the extended formulation that maps the extension polyhedron to
itself such that, for every vector y in the extended space, apply-
ing Tr to the projection of y yields the same vector as projecting the
vector obtained from y by applying Kr. Indeed, many extended for-
mulations are symmetric in a similar way, for instance the extended
formulation of the permutahedron by the Birkhoff-polytope men-
tioned in the Introduction as well as the extended formulation for
the spanning tree polytope discussed in Section 2.1.
In order to state Yannakakis' result more precisely, denote
by LMe(n) the set of all matching of cardinality f in the complete
graph with n nodes, and by p n- =- conv{x(M) : M e Mi(n)}
the associated polytope. In particular, j.._ '-i is the perfect-
matching-polytope (for even n).

Theorem 3 (Yannakakis [34]). For even n, the size of every symmet-
ric extended formulation of Pch(n) is at least (( [n 2 )).
1n/2 ((n-2 /4
Since P j (n) is (isomorphic to) a face of pmatch (n), one easily de-
rives the above mentioned exponential lower bound on the sizes of
symmetric extended formulations for pmath (n) from Theorem 3.
At the core of his beautiful proof of Theorem 3, Yannakakis shows
that, for even n, there is no symmetric extended formulation in
equation form (i.e., with equations and nonnegativity constraints
only) of Patch (n) of size at most (n) with k [(n 2)/4]. From
such a hypothetical extended formulation EFi, he first constructs an
extended formulation EF2 in equation form on variables YA for all
matching A with |A| < k such that the 0/I-vector valued map s*
on the vertices of Pnatch(n) defined by s*(x(M))A 1 if and only
if A c M is a section of EF2, i.e., s* (x) maps every vertex x to a

OPTIMA 85

preimage under the projection of EF2 that is contained in the ex-
tension polyhedron. Then it turns out that an extended formulation
like EF2 cannot exist. In fact, for an arbitrary partitioning of the node
set into two parts Vi and V2 with |Vi = 2k + 1, one can construct
a nonnegative point y* in the affine hull of the image of s* (thus
y* is contained in the extension polyhedron of EF2 that is defined
by equations and nonnegativity constraints only) with y* 0 for
all edges e connecting VI and V2, which implies that the projection
of the point y* violates the inequality x(6(Vi)) > 1 that is valid
for P .... (since Vi| 2k + 1 is odd). The crucial ingredient for
constructing EF2 from EFi is a theorem of Bocherts' [8] stating that
every subgroup G of permutations of m elements that is primitive
with |G| > m!/[(m + 1)/2]! contains all even permutations. Yan-
nakakis constructs a section s for EFi for that he can show by
exploiting Bochert's theorem that there is a nonnegative matrix C
with s(x(M)) = C s*(x(M)) for all M :Mn/2 (n), which makes it
rather straight forward to construct EF2 from EFi.
With respect to the fact that his proof yields an exponential lower
bound only for symmetric extended formulations, Yannakakis [34] re-
marked "we do not think that asymmetry helps much" in construct-
ing small extended formulations of the (perfect) matching polytopes
and stated as an open problem to "prove that the matching (...)
polytopes cannot be expressed by polynomial size LP's without the
symmetry assumption". As indicated above, today we still do not
know whether this is possible. However, at least it turned out re-
cently that requiring symmetry can make a big difference for the
smallest possible size of an extended formulation.

Theorem 4 ([22]). All symmetric extended formulations ofPlJ"ch (n)
have size at least n 1(logn), while there are polynomial size non-symmetric
extended formulations for Pmatc i (n) (i.e., the extension complexity
of Ptgnjh (n) is bounded from above by a polynomial in n).

Thus, at least when considering matching of size [log n] instead of
perfect (or arbitrary) matching, asymmetry indeed helps much.
While the proof of the lower bound on the sizes of symmetric
extended formulations stated in Theorem 4 is a modification of Yan-
nakakis' proof indicated above, the construction of the polynomial
size non-symmetric extended formulation of Pch (n) relies on
the principle of disjunctive programming (see Section 2.2). For an
arbitrary coloring ( of the n nodes of the complete graph with 2k
colors, we call a matching M (with |MI = k) (-colorful if, in each
of the 2k color classes, there is exactly one node that is an end-
node of one of the edges from M. Let us denote by PC the convex
hull of the characteristic vectors of (-colorful matching. The crucial
observation is that PC can be described by 0(2k + n2) inequalities
(as opposed to Q(2n) inequalities needed to describe the polytope
associated with all matching, see the Introduction). On the other
hand, according to a theorem due to Alon, Yuster, and Zwick [2],
there is a family of q such colorings (1,..., Cq with q 20(k) log n
such that, for every 2k-element subset W of the n nodes, in at least
one of the colorings the nodes from W receive pairwise different
colors. Thus we have P h (n) = conv(P1 u ... u PC), and hence
(as described in Section 2.2) we obtain an extended formulation
of match (n) of size 20(k) n2 log n, which, for k [log n], yields the
upper bound in Theorem 4.
Yannakakis [34] moreover deduced from Theorem 3 that there
are no polynomial size symmetric extended formulations for the
traveling salesman polytope (the convex hull of the characteristic
vectors of all cycles of lengths n in the complete graph with n
nodes). Similarly to Theorem 4, one can also prove that there are
no polynomial size symmetric extended formulations for the poly-
topes associated with cycles of length [log n], while these polytopes
nevertheless have polynomially bounded extension complexity [22].

Pashkovich [27] further extended Yannakakis' techniques in order
to prove that every symmetric extended formulation of the permu-
tahedron PPerm (n) has size at least 0 (n2), showing that the Birkhoff-
polytope essentially provides an optimal symmetric extension for the
permutahedron.

5 Conclusions
Many polytopes associated with combinatorial optimization prob-
lems can be represented in small, simple, and nice ways as projec-
tions of higher dimensional polyhedra. Moreover, though we have
not touched this topic here, sometimes such extended formulations
are also very helpful in deriving descriptions in the original spaces.
What we currently lack are on the one hand more techniques to
construct extended formulations and on the other hand a good un-
derstanding of the fundamental limits of such representations. For
instance, does every polynomially solvable combinatorial optimiza-
tion problem admit an extended formulation of polynomial size? We
even do not know this for the matching problem. How about the
stable set problem in perfect graphs? The best upper bound on the
extension complexity of these polytopes for graphs with n nodes
still is no(logn) (Yannakakis [34]).
Progress on such questions will eventually shed more light onto
the principle possibilities to express combinatorial problems by
means of linear constraints. Moreover, the search for extended for-
mulations yields new modelling ideas some of which may prove to
be useful also in practical contexts. In any case, work on extended
formulations can lead into fascinating mathematics.

Acknowldgements. We are grateful to Sam Burer, Kanstantsin
Pashkovich, Britta Peis, Laurence Wolsey, and Mihalis Yannakakis for
ducing Figure I.

Volker Kaibel, Institut for Mathematische Optimierung, Fakultit for Ma-
thematik, Otto-von-Guericke Universitit Magdeburg, Universititsplatz 2,
39106 Magdeburg, Germany. kaibel@ovgu.de

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Discussion Column

Laurence A. Wolsey

Using Extended Formulations in Practice

Though Dantzig-Wolfe decomposition [7] dating from 1960 and the
description of the convex hull of the union of polyhedra of Balas
[2] from the early 1970s can both be seen as results about using
additional variables in modeling, systematic interest in what we now
call extended formulations (EFs) seems to have begun in the 1980s.
The flurry of activity in polyhedral combinatorics in the 1970s con-
centrated on the development of valid inequalities to strengthen the
formulation of both easy and NP-hard integer programs, and only
later was the question of adding variables to obtain tighter formula-
tions raised systematically.
In this short discussion, we try to indicate a few of the areas in
which EFs have been used computationally, as well as the different
ways in which they have been tested. Two recent surveys [5] and
[22] contain many more examples of EFs and the techniques avail-
able for constructing such formulations, the first concentrating more
on combinatorial optimization problems and the second more on in-
teger programming. Note that we do not discuss the use of an EF to
generate valid inequalities in the original space of variables, as in the
case of Benders' algorithm [4] or lift-and-project [3].
Lot-sizing, network design and routing models are three of the
areas in which EFs have been effective computationally. Below we
will first describe one or two EFs in each of these areas. Then we
will briefly indicate different ways in which they have been used.

Lot-Sizing Direct Use of an MIP Solver
Lot-sizing problems provide perhaps the richest class of problems
for which EFs have been developed that are small in size and compu-
tationally effective. Two reasons for this are the fact that the single
item problem can be viewed as a fixed charge network flow prob-
lem and that the uncapacitated and constant capacity variants typi-
cally can be solved in polynomial time via dynamic programming. In
1977 Krarup and Bilde [13] published a reformulation of the single-
item uncapacitated lot-sizing problem as an uncapacitated facility lo-
cation problem and showed that for the specific objective function
obtained for lot-sizing, the linear programming relaxation had inte-
ger solutions. Another way to explain this reformulation is as a mul-
ticommodity reformulation of the initial fixed charge network flow
formulation in which the demand for each time period is treated as
a separate commodity. Recently such a multicommodity reformula-
tion has been shown to be effective computationally [15] for 2-level
production-transportation problems with on the order of 2-5 pro-
duction sites, 5-10 items, 10-40 clients and 12-24 time periods.
In an important paper in 1987 Eppen and Martin [8] showed that a
shortest path EF of the same single item uncapacitated problem was
actually an integer polytope. More generally they showed how in
many cases a dynamic programming algorithm for optimization over
a discrete set X leads through LP duality to an extended formula-
tion whose projection onto the original space of variables gives the
convex hull of solutions conv(X). They developed EFs for several dif-
ferent lot-sizing variants and among others solved to near-optimality
multi-item big bucket lot-sizing problems with up to 200 products
and 10 time periods.
Pochet and Wolsey [ 17] and Constantino [6] developed compact
and tight EFs for several variants of the constant capacity single-item
lot-sizing problem. Based on this work, Gunluk and Pochet [ 12] later

OPTIMA 85

showed that the important structure underlying many of these re-
sults was a simple mixed integer set of the form {(s,x) e R x Zn :
s + xt > bt, t = ...,n}, called the mixing set. The book [18]
describes an automatic reformulation procedure to enable the user
to benefit from these EFs and contains several production planning
case studies using them. The direct approach used for the computa-
tional results cited so far is to take the multi-item problem, add the
EFs for each item, and then feed the resulting formulation directly to
an MIP solver.

The Tree Polytope and Routing Using Approximate EFs
A well-known formulation for the set X of incidence vectors of
spanning trees in a graph consists of degree constraints and an ex-
ponential number of subtour elimination constraints using variables
x e R|E| where xe = 1 denoting that edge e is in the spanning
tree. Wong [23] proposed an EF for the symmetric traveling sales-
man problem that provides a tight formulation for the spanning tree
polytope. Specifically to model the connectivity of a spanning tree,
one chooses a root node r and then constructs an arborescence
rooted at r in which it is possible to direct a unit of flow from
the root to each node i # r. This leads to a formulation as a sin-
gle source fixed charge network flow problem on the graph (V, E):
{(x,y,f) {0,1}E x{0,1}AI R+ Xe yij+ jie (i,j)
E, j fij fji= -lir, fij < (V -1)yij (i,j) e A},where
Yij 1 if arc (i,j) is in the arborescence and fij is the flow in arc
(i, j). Again introducing a distinct commodity for each node k # r,
one obtains the EF {(x,y,w) e {0,1}1 E x {0,1}A x I .
Xe = yij + yji Ve (i,j) e E, 2j w iw 0 Vk, i
r,k, Yj wk 1 Vk r, w. if the directed path from the root to node k passes through the arc
(i,j). With the cardinality constraint YeExe Vi 1, this EF is
an alternative to that of Martin for the convex hull of incidence vec-
tors of spanning trees described above by Kaibel. This formulation
involves 0(n3) variables and constraints if n -VI which leads to
formulations that are too large for practical use if n exceeds 30-40.
to get one of more reasonable size. Specifically the idea is to drop
the k, i flow conservation constraint if node i is not within some
selected neighborhood of node k, as well as the variables w k wki if
neither i or j is in the neighborhood. This relaxation allows one to
solve traveling salesman problems with 70 or so nodes directly with
an MIP solver.

Multi-commodity Fixed Charge Network Flows Column and Row
Generation
Magnanti et al. [14] considered the following model for a single arc:
X {(x,y) e R, x Z, : _I dkXk < Cy, x < 1}, and showed
that the exponential family of Residual Capacity or MIR inequali-
ties: Y, ', 'j < ar + pTY where T c {1,...K}, p1 [ (T],
OT d(T) pC and OT (C fr)PT provide the convex hull.
Recently Frangioni and Gendron [9] have shown that the follow-
ing polyhedron Q provides an EF for conv(X): {(x,y,v, w) e
K x p x R x pKS : 1Sl Vs < 1, (S 1)CVs < ilkdkWks <
sCvs V s, Wks < Vs V k,s, y = Vs,Xk s= wks V k},
XK
where S [ 1 and we can interpret the additional variables
as follows: Vs 1 if y = s and vs = 0 otherwise, and Wks Xk
if y s and Wks 0 otherwise. Here the formulation obtained by
adding this EF for each arc is too large to be solved directly by an
MIP solver. The authors develop a column and row generation al-
gorithm which works with a subset of the variables and constraints,
solves an optimization problem over each single arc set to find miss-
the constraints in which these variables occur. The authors use the

approach to obtain strong lower bounds for multicommodity ca-
pacitated network design problems with up to 30 nodes and 400
commodities.

Parallel Machine Scheduling Cutting Planes
Given a set of n jobs, a natural first choice is the set of variables:
tj denoting the start time of job j. If the processing times of the
jobs are integer, the time-indexed variables [19] where wit 1 if
job j starts processing in time t is a common choice for additional
variables. These allow one to model many machine scheduling prob-
lems, but already the resulting extended formulations are too large
to be fed directly to an MIP solver. However surprisingly it can be of
interest to consider an even larger set of variables, such as zit = 1
if job i finishes and job j starts at time t on the same machine,
and job 0 corresponds to idle. With these variables, it is not diffi-
cult to see that the following holds for any subset S c N of jobs:
t i 51ies,jeS tzijt _- 1t iS',jes tijt jes Pj, where T is
an upper bound on the time horizon. Setting ut -= icses zijt
and vt = Zisjs zijt, this becomes the knapsack set {(u, v) e
Zx : 1 ut t=l t = jes Pj}, for which one can gen-
erate cutting planes. Note that these cannot in general be converted
into cutting planes in either the tj variables or in the Wjt variables.
In a recent paper of Pessoa et al. [ 16] such cutting planes form one of
the important computational steps of an algorithm allowing the solu-
tion of parallel machine scheduling problems with up to 4 machines
and 100 jobs. This idea appeared in a paper of Gouveia [II], and
has been used in several successful studies in vehicle routing, [10],
[20], etc. combining cutting planes in both the original and additional
variables and column generation, under the title branch-and-price-
and-cut.
We have not discussed in detail the size of the EFs presented
above. However if they are more than quadratic in the size of the
original formulation, it is typically not possible to solve them directly
with an MIP solver. In such cases considerable algorithmic original-
ity is required to successfully use EFs and this is an intriguing area
for research. Apart from the approaches discussed her, two more
standard directions include the use of EFs to develop heuristics, see
for instance [8], and to derive valid inequalities in the original space
(variants of Benders' algorithm).

Laurence A. Wolsey, Center for Operations Research and Economet-
rics (CORE), Universit6 catholique de Louvain, Voie du Roman Pays 34,
1348 Louvain-la-Neuve, Belgium. laurence.wolsey@uclouvain.be

References
[I] Tobias Achterberg, Thorsten Koch, and Andreas Tuchscherer. On the ef-
fects of minor changes in model formulations. Technical Report 08-29,
Zuse Institute Berlin, Takustr. 7, Berlin, 2009.
[2] E. Balas. Disjunctive programming: Properties of the con-
vex hull of feasible points. Invited paper with foreword by
G. Cornu6jols and W.R. Pulleyblank Discrete Applied Mathematics,
89:1-44, 1998.
[3] E. Balas, S. Ceria, and G. Cornuejols. A lift-and-project algorithm for mixed
0-1 programs. Mathematical Programming, 58:295-324, 1993.
[4] J.F Benders. Partitioning procedures for solving mixed variables program-
ming problems. Numerische Mathematik, 4:238-252, 1962.
[5] M. Conforti, G. Cornuejols, and G. Zambelli. Extended formulations in
combinatorial optimization. 40R: A Quarterly journal of Operations Research,
8:1-48, 2010.
[6] M. Constantino. Lower bounds in lot-sizing models: A polyhedral study.
Mathematics of Operations Research, 23:10 1-118, 1998.
[7] G.B. Dantzig and P. Wolfe. Decomposition principle for linear programs.
Operations Research, 8:101-1 I I, 1960.
[8] G.D. Eppen and R.K. Martin. Solving multi-item lot-sizing problems using
variable definition. Operations Research, 35:832-848, 1987.

April 201 I1

[9] A. Frangioni and B. Gendron. 0-1 reformulations of the multicommod-
ity capacitated network deign problem. Discrete Applied Mathematics,
157:1229-1241, 2009.
[10] R. Fukosawa, H. Longo, J. Lysgaard, M. Reis, E. Uchoa, and R.F Werneck.
Robust branch-and-cut-and-price for the capacitated vehicle routing prob-
lem. Mathematical Programming, 106:491-511, 2006.
[II] L. Gouveia. A 2n-constraint formulation for the capacitated minimal span-
ning tree problem. Operations Research, 43:130-141, 1995.
[12] 0. Gunlk and Y. Pochet. Mixing mixed integer inequalities. Mathematical
Programming, 90:429-457, 200 1.
[13] J. Krarup and 0. Bilde. Plant location, set covering and economic lot sizes:
An O(mn) algorithm for structured problems. In L. Collatz et al., edi-
tor, Optimierung bei Graphentheoretischen und Ganzzahligen Probleme, pages
155-180. Birkhauser Verlag, Basel, 1977.
[14] T.L. Magnanti, P. Mirchandani, and R. Vachani. The convex hull of two core
capacitated network design problems. Mathematical Programming, 60:233-
250, 1993.
[15] R. Melo and L.A. Wolsey. Mip formulations and heuristics for two-level
production/transportation problems. Technical report, CORE, Universite
catholique de Louvain, 201 I.
[16] A. Pessoa, E. Uchoa, M.P. de Aragao, and R. Rodrigues. Exact algorithm
over an arc-time-indexed formulation for parallel machine scheduling prob-
lems. Mathematical Programming Computation, 2:259-290, 2010.
[17] Y. Pochet and L.A. Wolsey. Lot-sizing with constant batches: Formula-
tion and valid inequalities. Mathematics of Operations Research, 18:767-785,
1993.
[18] Y. Pochet and L.A. Wolsey. Production Planning by Mixed Integer Programming.
Springer, 2006.
[19] A.A.B. Pritsker, L.J. Watters, and P.J. Wolfe. Multiproject scheduling with
limited resources: a zero-one programming approach. Management Science,
16:93-108, 1969.
[20] E. Uchoa, R. Fukasawa, J. Lysgaard, A. Pessoa, M.P. Aragao, and D. Andrade.
Robust branch-and-cut-and-price for the capacitated minimum spanning
tree problem over an extended formulation. Mathematical Programming,
112:563-591, 2004.
[21] M. Van Vyve and L.A. Wolsey. Approximate extended formulations. Math-
ematical Programming B, 105:501-522, 2006.
[22] F Vanderbeck and L.A. Wolsey. Reformulation and decomposition of in-
teger programs. In M. Junger et al., editor, 50 Years of Integer Programming
1958-2008, pages 431-502. Springer, 2010.
[23] R.T. Wong. Integer programming formulations of the traveling salesman
problem. Proceedings of 1980 IEEE International Conference on Circuits and
Computers, xx: 149-152, 1980.

Mihalis Yannakakis

On Extended LP Formulations

In connection with his paper, Volker Kaibel asked me to give some
background on my paper "Expressing Combinatorial Optimization
Problems by Linear Programs" [3] regarding the motivation and
thoughts that guided that work. That research was carried out in
1987 and was presented first at the STOC'88 conference. It was mo-
tivated on the one hand by a claimed proof of P=NP that appeared
at that time and attracted a lot of attention in the community, and
on the other hand by the developments in the preceding years in
Linear Programming and the polyhedral approach to combinatorial
optimization.
In 1986-87 E. R. Swart circulated a paper that claimed to solve the
Traveling Salesman Problem using Linear Programming [2]. In partic-
ular, the paper constructed a large LP with many extra variables (n8
in the original version, n10 in a revision), which purportedly pro-
vided an extended formulation of the TSP polytope. Given the size
and complexity of the LP, it was rather hard to determine what was
exactly the effect of these variables and constraints and whether the
LP worked or not. It was clear that some methodology was needed
to understand what is possible to achieve with extra variables and
whether this approach could possibly work in principle.

The approach itself is actually a reasonable one to try if one be-
lieves that P=NP. First, Linear Programming is a P-complete prob-
lem, so it is in some sense a hardest, universal problem in P. Sec-
ond, we know that the introduction of extra variables (which are
then existentially quantified, i.e. projected out) is a powerful tool in
various domains that can increase drastically the expressive power
of a model, turning an exponential object into a polynomial one.
There are for example simple polytopes with an exponential num-
ber of facets, that can be expressed succinctly with extra variables.
In logic, every Boolean predicate can be represented by an equiv-
alent Boolean formula (in conjunctive normal form), but the for-
mula requires often exponential size. However, introducing extra
variables we can express any NP predicate by a polynomial-size for-
mula; this is essentially Cook's theorem showing that Satisfiability is
N P-complete.
Another motivation came from the discovery of Khachian's el-
lipsoid algorithm and its applications, and Karmakar's algorithm.
Grotchel, Lovasz and Schrijver had shown that the ellispoid algo-
rithm could be used to solve in polynomial time various problems
(such as the clique and independent set problem on perfect graphs)
even though their standard LP formulation has an exponential num-
ber of constraints (provided there is a good separation algorithm for
the constraints). In view of the impracticality of the ellipsoid algo-
rithm, it would be desirable to use instead Karmakar's algorithm (or
Simplex) for these problems; however this would require a polyno-
mial size LP description, which raises the question whether one can
construct such a formulation using any (small) set of extra variables
and constraints. The same question is relevant also for NP-hard
problems, such as the TSP, with respect to useful classes of their
facets; presumably (if P NP) we cannot construct exact extended
formulations in polynomial time, but perhaps we can express suc-
cinctly important classes of facets that provide useful cutting planes.
With this background I tried to take a systematic look to get
some understanding of what can and cannot be achieved with com-
pact extended LP formulations. The paper resolved partially some
questions and left many more open. For general polytopes a pleas-
ant surprise was that the complexity of the smallest extended LP
formulation can be characterized by a concrete parameter, the non-
negative rank of the slack matrix. The problem is that it is often
not that easy to compute the nonnegative rank. The paper pointed
out a connection to communication complexity, which can be used
sometimes to obtain lower or upper bounds, and applied it to the
independent (stable) set polytope for some classes of graphs. In gen-
eral however, we still need to develop good methods to estimate or
bound the nonnegative rank.
With respect to the question of expressing the TSP polytope by
a small LP, I could not show that this is impossible, but could rule it
out at least for the class of symmetric LPs (which includes the pro-
posed LP in Swart's attempted proof): any such formulation must
have exponential size. It was easier to prove this result first for the
(perfect) matching polytope and then transfer it by reduction to the
TSP polytope. (The result holds for a somewhat larger class of LPs
but it was simpler to state it for symmetric LPs.) In subsequent years
there have been several more similar attempts to prove P=NP using
an extended LP formulation for the TSP; they are generally symmet-
ric or almost so (sometimes for example a particular node is singled
out as the starting node of the tour; the exponential lower bound
holds even if a constant fraction of nodes is singled out.) Symmetry
seems to be rather natural in the construction of exact LP formu-
lations, but as shown recently by Kaibel, Pashkovich and Theis [I],
nonsymmetry can save a superpolynomial amount in some cases.
It remains an intriguing open question whether the matching poly-
tope can be expressed by a polynomial-size (unrestricted) LP. For the

OPTIMA 85

TSP and the polytopes of other NP-hard problems, we expect that
this must be impossible. Is this tantamount to showing P NP? It
does not seem so. The P=NP question is equivalent to a related
but somewhat different question, reflecting in a sense the differ-
ence between decision and optimization problems: P=NP iff we can
construct efficiently a polynomial-size extended LP formulation of a
polyhedron that includes the characteristic vectors of Hamiltonian
graphs and excludes those of non-Hamiltonian. Such an LP could be
easily obtained from a compact LP formulation for the TSP poly-
tope, but the converse does not seem to hold in any obvious way.
I believe in fact that it should be possible to prove that there is no
polynomial-size formulation for the TSP polytope or any other NP-
hard problem, although of course showing this remains a challenging

Mihalis Yannakakis, Department of Computer Science, Columbia University.
mihalis@cs.columbia.edu

References
[I] V. Kaibel, K. Pashkovich, D.O. Theis, "Symmetry matters for the sizes of
extended formulations", Proc. 14th IPCO, pp. 135-148, 2010.
[2] E. R. Swart, "P=NP," Technical Report, University of Guelph, 1986; revision
1987.
[3] M. Yannakakis, "Expressing Combinatorial Optimization Problems by Linear
Programs," Computer System Sci. 43(3), pp. 441-466, 1991.

Call for nomination for the

2012 George B. Dantzig Prize

Nominations are solicited for the George B. Dantzig Prize, admin-
istered jointly by the Mathematical Optimization Society (MOS) and
the Society for Industrial and Applied Mathematics (SIAM). This
prize is awarded to one or more individuals for original research
which by its originality, breadth and depth, is having a major impact
on the field of mathematical optimization. The contributions) for
which the award is made must be publicly available and may belong
to any aspect of mathematical optimization in its broadest sense.
The prize will be presented at the 2012 International Symposium
on Mathematical Programming, to be held August 19-24, 2012, in
Berlin, Germany. The members of the prize committee are John
Birge (Chair), Gerard Cornuejols, Yuri Nesterov, and Eva Tardos.
Nominations should consist of a letter describing the nominee's
qualifications for the prize, and a current curriculum vitae of the
nominee including a list of publications. They should be sent to
John Birge
University of Chicago
5807 South Woodlawn Avenue
Chicago, IL 60637, USA
Email: John.Birge@ChicagoBooth.edu
and received by 15 November 201 I. Submission of nomination ma-
terials in electronic form is strongly encouraged.

Announcements

Mathematical Programming, Series A,

is going online

Starting in January, 2011 all submissions to Mathematical Program-
ming, Series A, should be made through Springer's Online Manuscript
Submission, Review and Tracking System for the journal, at the web site
www.editorialmanager.com/mapr/.
have such online submission systems, and the move to such a sys-
tem was long overdue for Mathematical Programming. There are many
advantages of such a system starting with the submission, the ed-
itorial progress of each paper is recorded, the author can check on
the status of his/her submission, automatic reminders are sent to
the editors if they are late, etc. Moreover, statistics on the workload
and performance of each editor are easy to obtain. One important
detail is that the online system, at least initially, will be for the use of
authors and the editorial board only all communications between
the editors and referees will remain on a personal level, outside of
the system.
We hope that this move will improve the overall performance of
the journal and will help to eliminate outlier cases where an author
waits in frustration for a long-overdue report on his/her paper. Of
course, as with any new system some tuning will probably be nec-
essary to deal with unforseen problems, but the initial testing phase
went smoothly. Once the online system is well established for Series
A of Mathematical Programming, it may also be adapted for the use of
Series B of the journal.
We believe that this change is an important step in the journal's
organization, and are confident that it will result in a more reliable
and professional service to the optimization community.
Kurt Anstreicher (MPA Editor-in-Chief)
Alexander Shapiro (Chair of the MOS Publications Committee)
Philippe Toint (Chair of the MOS)

Mixed Integer Programming 201 I

June 20-23, 2011, University of Waterloo, Canada

You are cordially invited to participate in the upcoming workshop
in Mixed Integer Programming (MIP 201 1). The 2011 Mixed Inte-
ger Programming workshop will be the eighth in a series of annual
workshops held in North America designed to bring the integer pro-
gramming community together to discuss very recent developments
in the field. The workshop is especially focused on providing oppor-
tunities for junior researchers to present their most recent work.
The workshop series consists of a single track of invited talks. MIP
2011 is scheduled immediately following IPCO XV, which will take
place at IBM T.J. Watson Research Center in Yorktown Heights, NY
from June 15-17 (http://ipco201 L.uai.cl).

Confirmed speakers o Amitabh Basu UC Davis o Gerard Cornuejols
- Carnegie Mellon University o Claudia D'Ambrosio University of
Bologna o Santanu Dey Georgia Tech o Sarah Drewes UC Berke-
ley o Samir Elhedhli University of Waterloo o Marcos Goycoolea
Mellon University o Adam Letchford Lancaster University o Leo
Liberti Ecole Polytechnique o Marco Luebbecke RWTH Aachen
University o Susan Margulies Rice University o Alex Martin Uni-
versitt Erlangen-Nrnberg o Giacomo Nannicini Carnegie Mellon
University o Michael Perregaard FICO o Sebastian Pokutta MIT
o Oleg Prokopyev University of Pittsburgh o Sebastian Sager Uni-
versity of Heidelberg o Domenico Salvagnin University of Padova
o Gautier Stauffer University of Bordeaux I o Laura Sanita Ecole
Polythechnique Federale de Lausanne o Levent Tuncel University
of Waterloo o Francois Vanderbeck University of Bordeaux I
o Robert Weismantel ETH Zurich

The workshop is designed to provide ample time for discussion and
interaction between the participants, as one of its aims is to facili-

April 201 I1

tate research collaboration. Thanks to the generous support by our
sponsors, registration is free, and travel support is available.
Program Committee: Shabbir Ahmed (Georgia Institute of Tech-
nology), Ricardo Fukasawa (University of Waterloo), Ted Ralphs
(Lehigh University), Juan Pablo Vielma (University of Pittsburgh), Gi-
acomo Zambelli (London School of Economics).
www.math.uwaterloo.ca/~mip201 I/

Optimization 2011

July 24-27, 2011, Lisbon (Caparica), Portugal, Department of Mathe-
matics, School of Sciences and Technology, New University of Lisbon

Optimization 201 I1 is the seventh edition of a series of Optimization
international conferences held every three or four years, in Portugal.
This meeting strives to bring together researchers and practitioners
from different areas and with distinct backgrounds, but with com-
mon interests in optimization. This conference series has interna-
tional recognition as an important forum of discussion and exchange
of ideas, being organized under the auspices of APDIO (the Por-
tuguese Operations Research Society).
In this edition, we feel honored to celebrate the 60th anniversary
of our dear colleague Joaquim Joao Judice (Univ. of Coimbra).
Confirmed plenary speakers: Gilbert Laporte (HEC Montreal),
Jean Bernard Lasserre (LAAS-CNRS, Toulouse), Jose Mario Martinez
(State University of Campinas), Mauricio G.C. Resende (AT&T Labs
- Research), Nick Sahinidis (Carnegie Mellon University), Stephen J.
Wright (University of Wisconsin).
We look forward to meeting you in Optimization 2011.
Ana Luisa Custodio (Co-chair of the Organizing Committee)
Paula Amaral (Co-chair of the Organizing Committee)
http://www.fct.unl.pt/optimization201 I

MOPTA 2011

August 17-19, 201 I, Lehigh University, Rauch Business Center, Bethle-
hem, PA, USA

MOPTA aims at bringing together a diverse group of people from
both discrete and continuous optimization, working on both theo-
retical and applied aspects. There will be a small number of invited
talks from distinguished speakers and contributed talks, spread over
three days.
Our target is to present a diverse set of exciting new devel-
opments from different optimization areas while at the same time
providing a setting which will allow increased interaction among
the participants. We strive to bring together researchers from
both the theoretical and applied communities who do not usually

have the chance to interact in the framework of a medium-scale
event.
Confirmed plenary speakers: Mark Daskin (U. of Michigan),
Michael Ferris (U. of Wisconsin), Adrian Lewis (Cornell U),
Jorge More (Argonne), Javier Pena (Carnegie Mellon), Cliff Stein
(Columbia U), Philippe Toint (U of Namur).
Organizing Committee: Katya Scheinberg (Chair), Tamas Terlaky,
Ted Ralphs, Robert Storer, Aurelie Thiele, Larry Snyder, Frank E.
Curtis.
We look forward to seeing you at MOPTA 201 I1.
http://coral.ie.lehigh.edu/~mopta/

OR 201 I

August 30-September 2, 2011, Zurich, Switzerland

The main goal of the conference is to bring together members of the
international OR community to discuss scientific progresses in vari-
ous subfields of OR in a truly interdisciplinary spirit. The highlights
and core of the conference are the invited Keynote Speakers and
the parallel semi-plenary lectures on various topics representing the
state of the art in these fields. Certainly, the conference provides a
platform to present current research and to compete for a publica-
tion in the referred proceedings.
Plenary lectures: Dimitris J. Bertsimas (MIT, Cambridge): Advances
in stochastic and adaptive optimization; Kenneth L. Judd (Hoover Insti-
tution, Stanford): Numerically Efficient and Stable Algorithms for Solv-
ing Large Dynamic Programming Problems in Economics, Finance, and
Climate Change Models; William Pulleyblank (United States Military
Academy, West Point, NY): Challenges and Opportunities for Opera-
tions Research in the next decade.
Program committee: Karl Schmedders (Chair, University of
Zurich), Friedrich Eisenbrand (EPF Lausanne), Luca Gambardella
(IDSIA, Lugano), Diethard Klatte (University of Zurich), Ulrike
Leopold-Wildburger (University of Graz), Hans-Jakob Luthi (ETH
Zurich), Stefan Nickel (Karlsruhe Institute of Technology), Stefan
Pickl (Universitat der Bundeswehr Munchen), Marion Rauner (Uni-
versity of Vienna), Brigitte Werners (Ruhr-Universitat Bochum)
Organization: The Swiss Association of Operations Research
(SVOR) is the premier organization in Switzerland for advancing the
profession, practice, and science of operations research (OR) and
management science (MS). Every four years the german speaking
OR societies from Austria (OGOR), Germany (GOR) and Switzer-
land (SVOR) organize a joint international conference OR 2011.
The local organizers are the Institute for Operations Research
(IFOR, Prof. H.-J. Luthi) of the ETH Zurich and the Institute for OR
(IOR, Prof. K. Schmedders) of the University of Zurich who under
the patronage of SVOR will share the organizational, financial and
scientific responsibilities.
http://www.or20 I I.ch

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.

OPTIMA 85

IPCO 2011

The I 5th Conference on Integer Programming and Combinatorial Optimization

IPCO XV will be held on June 15-17, 2011 at the IBM T.J. Watson
Research Center in Yorktown Heights, New York, USA.
Accepted papers (in order of submission) o Amitabh Basu, Gerard Cornuejols
and Marco Molinaro. A probabilistic analysis of the strength of the split and
triangle closures o Alexander Ageev, Yohann Benchetrit, Andras Sebo and
Zoltan Szigeti. An Excluded Minor Characterization of Seymour Graphs
o Stephan Held, Edward C. Sewell and William Cook. Safe Lower Bounds
For Graph Coloring o Mathieu Van Vyve. Fixed-charge transportation on
a path: Linear programming formulations o Britta Peis and Andreas Wiese.
Universal packet routing with arbitrary bandwidths and transit times o San-
tanu S. Dey and Sebastian Pokutta. Design and Verify: A New Scheme for
Generating Cutting-Planes o Jose A. Soto and Claudio Telha. Jump Number
of Two-Directional Orthogonal Ray Graphs o Anna Karlin, Claire Math-
ieu and Thach Nguyen. Integrality Gaps of Linear and Semi-definite Pro-
gramming Relaxations for Knapsack o Daniel Dadush, Santanu S. Dey and
Juan Pablo Vielma. On the Chvatal-Gomory Closure of a Compact Convex
Set o Deeparnab Chakrabarty, Chandra Chekuri, Sanjeev Khanna and Ni-
tish Korula. Approximability of Capacitated Network Design o Deeparnab
Chakrabarty and Chaitanya Swamy. Facility Location with Client Latencies
o Satoru Iwata and Mizuyo Takamatsu. Computing the Maximum Degree of
Minors in Mixed Polynomial Matrices via Combinatorial Relaxation o Pierre
Bonami. Lift-and-Project Cuts for Mixed Integer Convex Programs o Fab-
rizio Grandoni and Thomas Rothvoss. Approximation Algorithms for Single
and Multi-Commodity Connected Facility Location o Kenjiro Takazawa. Dis-
crete convexity and faster algorithms for weighted matching forests o Tamas
Kiraly and Lap Chi Lau. Degree Bounded Forest Covering o Monia Gian-
domenico, Adam Letchford, Fabrizio Rossi and Stefano Smriglio. A New
Approach to the Stable Set Problem Based on Ellipsoids o Aman Dhesi,
Pranav Gupta, Amit Kumar, Gyana Parija and Sambuddha Roy. Contact
Center Scheduling with Strict Resource Requirements o Volker Kaibel and
Kanstantsin Pashkovich. Constructing Extended Formulations from Reflec-
tion Relations o Sylvia Boyd, Rene Sitters, Suzanne van der Ster and Leen

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Stougie. TSP on Cubic and Subcubic Graphs o Yu Hin Au and Levent Tuncel.
Complexity Analyses of Bienstock-Zuckerberg and Lasserre Relaxations
on the Matching and Stable Set Polytopes o Mario Ruthmair and Gunther
Raidl. A Layered Graph Model and an Adaptive Layers Framework to Solve
Delay-Constrained Minimum Tree Problems o Inge Gortz, Marco Moli-
naro, Viswanath Nagarajan and R Ravi. Capacitated Vehicle Routing with
Non-Uniform Speeds o S. Thomas McCormick and Britta Peis. A primal-
dual algorithm for weighted abstract cut packing o Friedrich Eisenbrand,
Naonori Kakimura, Thomas Rothvoss and Laura Sanita. Set Covering with
Ordered Replacement Additive and Multiplicative Gaps o Oliver Fried-
mann. A subexponential lower bound for Zadeh's pivoting rule for solv-
ing linear programs and games o Claudia D'Ambrosio, Jeff Linderoth and
James Luedtke. Valid Inequalities for the Pooling Problem with Binary Vari-
ables o Bissan Ghaddar, Juan Vera and Miguel Anjos. An Iterative Scheme
for Valid Polynomial Inequality Generation in Binary Polynomial Program-
ming o Martin Bergner, Alberto Caprara, Fabio Furini, Marco LLibbecke,
Enrico Malaguti and Emiliano Traversi. Partial Convexification of General
MIPs by Dantzig-Wolfe Reformulation o Trang Nguyen, Mohit Tawarmalani
and Jean-Philippe Richard. Convexification Techniques for Linear Comple-
mentarity Constraints o William Cook, Thorsten Koch, Daniel Steffy and
Kati Wolter. An exact rational mixed-integer programming solver o Ojas
Parekh. Iterative packing for demand matching and sparse packing o Matteo
Fischetti and Michele Monaci. Backdoor branching
Program committee: Nikhil Bansal (IBM), Michele Conforti (Padova),
Bertrand Guenn (Waterloo), Oktay GunlUk (IBM), Tibor Jordan
(ELTE Budapest), Jochen Koenemann (Waterloo), Andrea Lodi
(Bologna), Franz Rendl (Klagenfurt), Giovanni Rinaldi (Roma), Gun-
ter Rote (FU Berlin), Cliff Stein (Columbia), Frank Vallentin (Delft),
Jens Vygen (Bonn), Gerhard Woeginger (Eindhoven, chair).
Organizing committee: Sanjeeb Dash, Oktay Gunlik (chair), Jon Lee,
Maxim Sviridenko.

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