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Mathematical Programming Society Newsletter

Steve Wright
MPS Chair's Column

March 16, 2010. You should recently have received a letter con-
cerning a possible change of name for MPS, to "Mathematical Op-
timization Society". This issue has been discussed in earnest since
ISMP 2009, where it was raised at the Council and Business meet-
ings. Some of you were kind enough to send me your views follow-
ing the mention in my column in Optima 80. Many (including me)
believe that the term "optimization" is more widely recognized and
better understood as an appellation for our field than the current
name, both among those working in the area and our colleagues
in other disciplines. Others believe that the current name should
be retained, as it has the important benefits of tradition, branding,
and name recognition. To ensure archival continuity in the literature,
the main titles of our journals Mathematical Programming, Series A
and B and Mathematical Programming Computation would not be
affected by the proposed change; only their subtitles would change.
As the letter explains, there will be a final period for comment on
the proposed change, during which you are encouraged to write to
me, to incoming chair Philippe Toint, or to other Council or Execu-
tive Committee members to express your views. Following discus-
sion of membership feedback, Council will vote on a motion to put
the name change (which involves a constitutional change) to a vote
of the full membership. If Council approves, you will then be asked
to vote on the proposal in the same manner as in the last election
for officers that is, by online ballot, with an option for a paper bal-
lot if you prefer. The proposal will pass if votes in favor exceed votes
against on the membership ballot. We anticipate that the matter will
be settled by May. Results will be announced by mail and on the web
site www.mathprog.org.
We have been greatly saddened by the loss of life in earthquakes
around the world in these early months of 2010. Fortunately, our
colleagues in Chile were not affected severely by the earthquake
of February 27 in that country. The MPS-organized conference IC-
COPT 2010 will take place as planned in Santiago, Chile during July
26-29, preceded by a School for graduate students and young re-
searchers on July 24-25. Conference facilities were left unscathed by
the earthquake, and the airport and local transportation networks
are operating normally. The local organizing committee (headed by
Alejandro Jofre) is making a great effort to ensure a successful con-
ference. I urge you to attend and contribute, and help add another
chapter to the short but illustrious history of ICCOPT.
Many of us will be making plans for other mid-year conferences,
and there are many to choose from in 2010. Our web site lists up-
coming meetings at http://www.mathprog.org/?nav=meetings, includ-
ing the MPS-organized IPCO 2010 (Lausanne, June 9-11) and other

MPS-sponsored meetings: the International Conference on Stochas-
tic Programming (ICSP) XII (Halifax, August 14-20), the Interna-
tional Conference on Engineering Optimization (Lisbon, September
6-9), and the IMA Conference on Numerical Linear Algebra and
Optimization (Birmingham, September 13-15).

Note from the Editors

The topic of this issue of Optima is Mechanism Design a Nobel
prize winning theoretical field of economics.
We present the main article by Jay Sethuraman, which introduces
the main concepts and existence results for some of the models
arising in mechanism design theory. The discussion column by Garud
lyengar and Anuj Kumar address a specific example of such a model
which can be solved by the means of optimization.
Optima 82 publishes the obituary of Paul Tseng by his friends
and colleagues Dimitri Bertsekas and Tom Luo, we are honored to
remember the contribution of this distinguished colleague.
The issue is also filled with announcements and advertisements.
Among them, we like to point out the announcement of the re-
cently published book about the first 50 years of Integer Program-
ming based on the commemoration of the seminal work of Gomory
which was held in Aussois as part of the 12th Combinatorial Opti-
mization Workshop in 2008. Announcing such a book, we get the
chance of correcting a mistake in the printed version of Optima 76
in which an article by Jon Lee about that workshop was published
without Jon's name. We apologize for that.

Contents of Issue 82 / April 2010
I Steve Wright, MPS Chair's Column
2 Jay Sethuraman, Mechanism Design for
House Allocation Problems: A Short Introduction
8 Garud lyengar and Anuj Kumar,
Parametric Network Flows in Adword Auctions
10 Dimitri P. Bertsekas and Zhi-Quan (Tom) Luo,
Paul Tseng, 1959-2009
12 No-Charge Access to
IBM ILOG Optimization Products for Academics
12 MIPLIB2010 Call for contribution
12 Conferences
14 SIAG/Optimization Prize: Call for Nominations
15 Imprint


Jay Sethuraman

Mechanism Design for House Allocation

Problems: A Short Introduction

I Introduction
How can a group of agents make a collective choice when not
all of the relevant information is commonly known to them? Hur-
wicz [16, 17] provided the first mathematical formulation of this
problem. In this formulation, participants exchange messages, and
the mechanism itself is thought of as a black-box that determines an
outcome as a function of all the messages exchanged. Each agent is,
of course, strategic in the sense that he may choose not to commu-
nicate certain information if he believes that doing so may result in
an outcome that he prefers less. In this environment, what sorts of
outcomes can be implemented as equilibria of these message games?
This is the central question that the area of mechanism design is con-
cerned with. There are a number of recent surveys that offer an
accessible overview of the area; especially recommended are the
ones by Jackson [19], and the wealth of material available on the
Nobel Prize website [26] that discusses the contributions of Hur-
wicz, Maskin, and Myerson, who were awarded the 2007 Economics
Prize for "having laid the foundations of mechanism design theory."
(The recent book by Nisan et al. [25] covers a lot of ground, and, in
particular, has two chapters on mechanism design theory.)
In this short article, we study the mechanism design question for
a particular class of problems that have come to be known as house
allocation problems. These form a special class of mechanism design
problems that are relatively well-understood. We start, however,
with the Gibbard-Satterthwaite theorem, which shows that the only
mechanisms that can be implemented truthfully in dominant strate-
gies are dictatorships.

2 The Gibbard-Satterthwaite Theorem
Suppose we are given a set of alternatives A with |A| > 3 and a set
of agents N with NI = n > 2. Let E be the set of all permutations
of A, and let oai c be agent i's preference over the alternatives.
The vector p = (ail,2,...,. n) is called a preference profile. A
social choice function f selects an alternative for each preference
profile. Formally, it is a function f : Yn A. While this description
is reminiscent of a voting problem the alternatives are candidates
in an election, the agents are the voters, and the chosen alternative
is the winner it is general enough to model a wide variety of situa-
tions. Nevertheless, the voting problem is a useful example to keep
in mind.
There are, of course, many social choice functions, but for prac-
tical purposes we are interested in social choice functions satisfying
some reasonable properties. Motivated by the voting example, we
can demand the following two properties of social choice functions:
- Unanimity: If a c A is the top choice of each agent in a profile
p, then f(p) = a.
- Monotonicity: Suppose f( l,ao2,... ,aon) = a. Suppose for
each agent i, ai' is such that xa~ja only if x~ia. Then
f(a;,a ...,an) = a.
We say that a social choice function f is dictatorial if there is an
agent i such that f selects a whenever i ranks a as his top choice.
The following theorem is now easy to prove.

Theorem I. Suppose there are exactly two agents. Then the only social
choice functions that satisfy unanimity and monotonicity are dictatorial.

Proof. Consider the profile p in which agent I's preference order-
ing is (a,b,...) and agent 2's preference ordering is (b, a,...). We
first argue that f must select a or b for profile p. Suppose other-
wise, and that f(p) c 1 a, b (such an alternative exists because
A| > 3). Consider what f chooses when agent I changes his pref-
erence ordering to (b, a,...): Unanimity dictates that f choose b,
whereas monotonicity dictates that f continue to choose c (as c's
relative ordering with respect to the other alternatives has not been
altered). So f(p) a or f(p) b. In fact this argument shows
that on any profile, the outcome chosen should be Pareto optimal: it
should be the first choice of agent I or of agent 2. Without loss of
generality, assume that f chooses a. We argue next that agent I is
a dictator.
As before, pick an alternative c 1 a, b. Consider the profile q
in which agent I has the preference ordering (c, a, b,...) and agent
2 has the ordering (b,c,a,...). We prove that f(q) c. By the
argument in the previous paragraph, f(q) is either b or c. But if
f(q) b, then monotonicity implies that f(p) b as well, con-
trary to our assumption that f(p) a. So f(q) c. Thus, by
monotonicity, f(r) = c for any preference profile r in which agent
I has c as his top-choice and agent 2 has a preference ordering
(b,c,...). Now, consider the profile s in which agent I has a prefer-
ence ordering (c, b, a,...) and agent 2 has an ordering (b, a..., c).
In particular, agent 2 ranks c as his last choice. By Pareto optimal-
ity, f(s) = c or f(s) = b. However, if f(s) = b, then monotonicity
implies that f (r) = b, a contradiction. Thus f(s) = c. But by mono-
tonicity, f(s) = c for any profile in which agent I ranks c first. But c
was an arbitrarily chosen alternative, so our argument shows that f
must choose the alternative that appears as agent I's top choice. D

Theorem I can be extended to the case of more than two agents,
the main idea being the following: given any non-dictatorial social
choice function fn for an n agent problem we can construct a so-
cial choice function fn 1, for (n 1) agents as follows: choose a
a C, and let f- l(p) = fn(p,a), for all pe 1-1. It is not hard
to show that for any n > 2, if fn is non-dictatorial, unanimous, and
monotonic, then so is fn1.
To summarize, we have shown that as long as there are at least
2 agents and at least 3 alternatives, the only social choice func-
tions that satisfy unanimity and monotonicity are dictatorships. One
can now ask: what do these properties mean? And why are the
relevant to mechanism design? Consider a situation in which f is
known to the agents, so that any agent i can evaluate the out-
come with the knowledge of the preferences of the other agents. Let
(i = (a(T, a2, a72 ., oi-1, oi+1, ... n) denote the preference profiles
of all the agents other than i. If the social choice function f is such
that agent i (weakly) prefers the outcome f(ai, a-i) to f(aoi, -i),
for every a e 1, (Ti e n1 1, then f is said to be strategyproof
Thus, if f is a strategyproof social choice function, then the partic-
ipating agents do not have an incentive to lie about their true pref-
erences to the mechanism designer, regardless of the preferences of
the other agents. The following result relates strategyproofness to
monotonicity and unanimity.
Proposition I. Any strategyproof social choice function f : In A
that is onto satisfies unanimity and monotonicity.
An immediate implication of Theorem I and Proposition I is the
following result, due to Gibbard [14] and Satterthwaite [34].
Theorem 2. Any strategyproof social choice function f : In A that
is onto is a dictatorship.
The Gibbard-Satterthwaite theorem identifies a set of properties of
social choice functions that are mutually incompatible. On the face of
it, this is a negative result that imposes severe restrictions on which

April 2010

kinds of social choice functions can be implemented. A closer exam-
ination of the result, however, suggests that the impossibility result
can be overcome in a number of ways, by weakening or getting rid
of some of the assumptions on which the theorem rests. In partic-
ular, the Gibbard-Satterthwaite result assumes that the preference
domain Y from which the agents draw their preferences is universal,
meaning that all orderings of the alternatives are permitted. There
are many settings, however, in which the natural preference domain
is much smaller. In the rest of this article we examine one such

3 House Allocation: Deterministic Mechanisms
3.1 Housing Markets
There is a large (and growing) literature on house allocation prob-
lems, originating from the early work of Shapley and Scarf [35], who
considered the following model. Suppose there are n agents, each of
whom owns a distinct house. Each agent has a strict preference or-
dering of all the houses (including his own). How should the agents
reallocate the houses amongst themselves?
To relate this question to our earlier discussion, we should specify
N and A, the set of agents and alternatives, respectively. Clearly, N
is simply the set of agents, but A, the set of alternatives, includes all
possible matching of agents to houses. Thus, |A| n!, and the uni-
versal preference domain would include all possible (strict and weak)
orderings of these alternatives. However, there are many applica-
tions in which the preference of agent i is sensitive only to i's own
assignment, and is insensitive to how the other houses are assigned
to the other agents. In particular, it is reasonable to look at the
model in which the agent preferences over individual houses is ex-
tended to agent preferences over assignments as follows: i (weakly)
prefers the matching p to the matching p' if i (weakly) prefers his
assignment under p to his assignment under p'. As it turns out,
under this definition of agent preferences, one can overcome the
Gibbard-Satterthwaite Theorem:

Theorem 3. For the house allocation problem with strict preferences,
there is a non-dictatorial, strategyproof social choice function f onto A,
the set of all matching of houses to agents.

Proof. The proof is constructive and builds a matching for any given
profile of strict preferences (of the agents over individual houses).
Construct a graph in which there is a node for each remaining agent;
there is an arc from node i to node j if agent i's top choice among
the remaining houses is the one owned by agent j. As there are
finitely many agents, and each node has an out-degree of I, there
must be a cycle. Agents in the cycle exchange their houses in the ob-
vious way: if (i, j, k,..., -) is the cycle, then i is assigned the house
that j owns, j is assigned the house that k owns, etc. Note that
each of the agents in the cycle gets their top choice. The agents
involved in the cycle (and hence the houses they collectively own)
are removed from the problem, and the same algorithm is applied
to the reduced problem. Observe that the reduced problem has the
same structure as the original problem: each house in the reduced
problem is owned by a distinct agent in the reduced problem. D

Let f(.) be the outcome of this algorithm on any reported profile
of agent preferences. Observe that f(.) is individually rational: each
agent's assignment is at least as good as the house he owned. We
show that f(.) is strategyproof. Suppose for some profile of pref-
erences (al, a2,..., an), agent i drops out of the problem in the
kth iteration of the algorithm, assuming ,i is his true preference.
Notice that the only objects that i strictly prefers to the one that

he's assigned to are those that belong to agents who depart sooner.
But none of these agents can be made to point to i by any manip-
ulation on the part of agent i, as i's preference report only affects
the agents he points to, not the agents who point to him. Thus, f(.)
is strategyproof. Furthermore f(.) is onto: given any matching p of
the houses to agents, construct a preference profile for the agents
in which p(i) is i's top choice; on this preference profile, clearly,
the algorithm will return p as the final allocation. Finally, it is clear
that f(.) is not dictatorial: one can easily construct preference pro-
files in which any given agent is not assigned his top choice under
f(.). O

The algorithm just described is attributed to Gale and is called the
top-trading cycles (TTC) algorithm. Note that the house allocation
problem just described is an instance of an exchange economy with
indivisible objects, and there are two standard solution concepts -
the competitive equilibrium and the core one can associate with
such a problem. Given a matching p of the houses to the agents,
and a price vector p = (pi, P2,..., Pn), with pi indicating the price
of the house owned by agent i, we say that (p, p) is a competitive
equilibrium if (i) p(i) < pi and (ii) pj > pi, if i prefers the house
owned by j to the house p(i). The first condition ensures that each
agent can afford the house he is assigned to, and the second con-
dition ensures that no better house is affordable. A matching p is
in the core if no coalition S of agents can do better by reallocating
the houses they own amongst themselves (where, by doing better,
we mean that each agent in S gets a house that he likes at least
as much, and at least one agent in S gets a house that he strictly
The description of the TTC algorithm suggests a very simple con-
struction of the price vector: all the houses that are eliminated
in iteration k of the algorithm have a price of n k. It is a sim-
ple matter to verify that this vector of prices supports the TTC
outcome as a competitive equilibrium. Furthermore, a simple in-
ductive argument shows that the core contains a unique match-
ing, which is the one found by the TTC algorithm! Shapley and
Scarf [35] originally proved the existence of a core matching (and
that it can be supported as a competitive equilibrium) using more
complicated machinery. Later in the paper they describe Gale's TTC
algorithm and show how it, too, finds a core matching that can be
supported as a competitive equilibrium. The uniqueness of the core
and the strategyproofness of the core mechanism were proved by
Roth & Postlewaite [3 1] and Roth [30] respectively. Later, Ma [22]
proved that the TTC mechanism is characterized by the require-
ments of strategyproofness, Pareto-efficiency, and individual rationality:
Pareto-efficiency is simply the core condition for the grand coali-
tion of all agents, whereas individual rationality is the core condition
applied to singleton agents.

3.2 House Allocation
A closely related class of models, first studied by Hylland and Zeck-
hauser [18], is called the house allocation problem. In this model
there are n agents, each with strict preferences over n indivisible
objects. The objective is to find a matching of the agents to the ob-
jects. As before, we focus on direct mechanisms in which the agents
report their preference orderings to the mechanism, which finds an
assignment for each agent. In contrast to the housing markets, where
the initial property rights play a prominent role, it is not immediately
clear what properties to demand of the mechanism: for example, if
the n objects are owned by the n agents collectively, so that each
agent has an equal claim to each object, how should the final allo-
cation be done? Of course, much depends on the preferences of
the agents if the top choices of the agents are all distinct objects,


efficiency would imply that each agent be given their best object.
Difficulties arise when multiple agents have the same top choice. In
particular, what should the assignment be when everyone ranks the
same object as their top choice?
Suppose only deterministic mechanisms are permitted, so that any
given preference profile must be mapped to a particular allocation of
agents to profiles. Suppose also that preferences are strict. The class
of priority mechanisms are prominent: start with a given ordering of
the agents (the ordering does not depend on reported preferences),
and let the agents choose their best available objects according to
this ordering. Clearly, the resulting allocation is Pareto efficient, and
the mechanism is coalitionally strategyproof (immune to joint ma-
nipulation by an arbitrary coalition of agents). It also satisfies a prop-
erty called reallocation-proofness, which simply means that no pair of
agents have an incentive to misreport their preferences, even if they
are allowed to exchange the objects that they are finally assigned to.
However, the priority mechanism is essentially dictatorial: the alloca-
tion chosen is always among the top-choices of the first agent in this
ordering. However, the class of mechanisms satisfying all of these
properties is far more general and includes mechanisms other than
priority mechanisms. As an example, associate with each object a
priority ranking of the agents (which is also fixed exogenously). The
TTC algorithm can be generalized to this "two-sided" model, assum-
ing that each object is owned by the agent at the top of its priority
list one difference here is that in the standard TTC algorithm each
agent owns exactly one object, whereas here some agents may own
more objects and some none. Moreover, once some objects and
agents are removed from the problem, those agents are removed
from the priority lists of the remaining objects as well. This class of
mechanisms was introduced by Abdulkadiroglu and Sonmez [4], and
it is not difficult to show that these mechanisms are reallocation-
proof, coalitionally strategyproof, and Pareto efficient. In fact, one
can generalize this class of mechanisms even further: rather than fix
a priority ordering for each object exogenously, one can fix an inher-
itance function that, at each stage, determines the ownership of each
object as a function of the partial allocation of the objects to the
agents at that stage. The only requirement of an inheritance func-
tion is that if an agent i owns an object a at a certain stage, then a
is continued to be owned by i as long as i remains in the problem.
The key result of Papai [27] is that every coalitionally strategyproof,
reallocation-proof, Pareto efficient mechanism is a TTC mechanism
with an inheritance function, and vice-versa. In recent work, Pycia
and Unver [29] characterize all coalitionally strategyproof, Pareto
efficient mechanisms and observe that this is a superset of the TTC
mechanisms with inheritance functions.
We end our brief discussion of deterministic mechanisms with
a couple of axiomatic characterizations. A mechanism is said to be
neutral if it is insensitive to the relabeling of the houses: if Tr is any
permutation of the houses, and every object a is replaced by 7r(a)
in all preference lists, then the mechanism assigns an object a to
agent i originally if and only if it assigns rr(a) to agent i in the re-
labeled problem. Another useful property is consistency, which re-
lates the allocation of the mechanism in the original problem to its
allocation in a reduced problem. Suppose for a given problem the
mechanism determines an allocation in which agent i receives the
object j(i). Suppose agent j and object p(j) are no longer in the
problem, so that the mechanism is applied to the reduced prefer-
ence lists in which p(j) does not appear in any preference list and
agent j is absent. The mechanism is consistent if it allocates p(i) to
each of the remaining agents i. Svensson [38] showed that every
coalitionally strategyproof and neutral mechanism is a priority mech-
anism; Ergin [I I] showed that every Pareto efficient mechanism that
is consistent and neutral is a priority mechanism.

4 House Allocation: Probabilistic Mechanisms
There are significant disadvantages to relying only on determinis-
tic mechanisms for house allocation. For example, if all the agents
have the same preference ordering of the objects, every determin-
istic mechanism must favor some agents over the others, and this
can be viewed as unfair. One way to restore fairness is to allow for
money, but this may not be appropriate in all situations: concrete
examples include organ exchanges and public school assignments,
but there are several others. In these markets, it is important to
find fair allocations, but without using money, necessitating the use
of randomization in the mechanism. Randomized mechanisms have
been extensively studied for house allocation problems, and some-
what less so for the Shapley-Scarf housing market (as here, the TTC
outcome is compelling).
Consider again the problem of fair allocation of a number of in-
divisible objects to a number of agents, each desiring at most one
object. If there is only one object, there is really only one fair and ef-
ficient solution: assign the object to an agent chosen uniformly at
random among the n agents. When there are many objects the
problem becomes substantially more interesting for a number of rea-
sons: (i) many definitions of fairness and efficiency are possible; (ii)
richer mechanisms emerge; and (iii) since preferences of the agents
over the objects have to be solicited from the agents, truthfulness of
the allocation mechanism becomes important. Under a probabilistic
mechanism each agent's allocation is described by a vector, with the
jth component indicating the probability that this agent is assigned
object j. As not every pair of vectors can be compared, strate-
gyproofness (and other properties) can be defined in many ways, of
which we consider two. A mechanism is said to be strategyproof if
the allocation under truthful reporting of preferences dominates the
allocation obtained by any other preference report, where the dom-
ination is in the sense of first-order stochastic dominance. In other
words, a mechanism is strategyproof if the probability of receiving
one of the k best objects is maximized when the agent reports his
preference ordering truthfully, for every k. A mechanism is said to
be weakly strategyproof if it is not possible to obtain an allocation
that dominates the allocation under truthful reporting.
A natural idea is to use a priority mechanism, with the priority
ordering chosen uniformly at random, i.e., every ordering of the
agents is equally likely. This is the random priority (RP) mechanism,
formally first studied by Abdulkadiroglu and Sonmez [2], and has a
number of attractive features: it is efficient (every assignment cho-
sen with a positive probability is Pareto efficient strategyproof, and
treats equals equally: agents with identical preferences are treated in
an identical manner, a priori). This may seem like the last word on
the problem, but it turns out that there are other compelling ran-
domized mechanisms.
To illustrate RP, consider the following example, due to Bogo-
molnaia and Moulin [7]. Suppose there are 4 agents 1,2, 3,4 and 4
objects a, b, c, d. Consider the preference profile in which agents I
and 2 rank the objects a > b > c > d and agents 3 and 4 rank the
objects b > a > d > c. The probabilistic assignment computed by
the RP mechanism is
a b c d

1 5/12 1/12 5/12 1/12
2 5/12 1/12 5/12 1/12
3 1/12 5/12 1/12 5/12
4 1/12 5/12 1/12 5/12
For example, agent I gets assigned object a whenever I is ranked
first, or whenever I is ranked second, and 2 is not first. The former
event occurs with probability 1/4 and the latter with probability 1/6,
which explains the first entry of the matrix.

April 2010

A second natural mechanism, proposed by Hylland and Zeck-
hauser [18], is to adapt the competitive equilibrium with equal in-
comes (CEEI): endow each agent with a dollar, and find a price for
each house so that the market clears: given the price vector, each
agent consumes a bundle (fractions of each house) that maximizes
his total utility, while staying within his budget. A standard fixed-point
argument shows the existence of market-clearing prices. Moreover
the (probabilistic) allocation so found is envy-free (a stronger prop-
erty than equal treatment of equals) and efficient (in a stronger sense
than Pareto efficiency). Efficiency follows by definition, and envy-
freeness follows from the fact that every agent has the same budget,
and so can afford every other agent's allocation at the current price.
But there are two potential objections to this solution: it can be
shown that this mechanism is not strategyproof; and furthermore,
the computational and informational requirements of implementing
this mechanism are prohibitive: it requires the solution of a fixed-
point problem, and requires a complete knowledge of the utility
functions of the agents. In contrast, the random priority mechanism
is very simple to implement, and only requires the agents to rank
the objects.
A third class of mechanisms, due to Bogomolnaia and Moulin [7],
is the probabilistic serial (PS) mechanism, which combines the attrac-
tive features of RP and CEEI: it requires the agents to report their
preferences over objects, not the complete utility functions, and yet
computes a random assignment that is envy-free and ordinally efficient.
Ordinal efficiency is stronger than the Pareto efficiency satisfied by
RP, but weaker than ex ante efficiency of CEEI, but given the "ordi-
nal" nature of the input to the mechanism (only preference rankings
are used, not complete utility functions), this is perhaps the most
meaningful notion of efficiency for an ordinal mechanism. The PS
mechanism can be motivated by the example we discussed earlier.
In that example, agents I and 2 both prefer a to b, whereas agents
3 and 4 prefer b to a. Yet, the RP mechanism assigns a with positive
probability to agents 3 and 4, and assigns b with positive probability
to agents I and 2, creating a potential inefficiency. As an alternative,
consider a mechanism that assigns a with probability 1/2 to agents I
and 2, and assigns b with probability 1/2 to agents 3 and 4; likewise
for objects c and d. The resulting allocation matrix

a b c d

1 1/2
2 1/2
3 0
4 0

1/2 0
1/2 0
0 1/2
0 1/2

is one in which every agent's allocation stochastically dominates (in
the first-order sense) his allocation under RP! It is this sense in which
RP is inefficient. The PS mechanism corrects for this inefficiency, and
can be described as follows: imagine that each agent eats his best
available object at each point in time at unit speed. Once one unit
of an object is consumed, it is removed from the problem. On this
example, initially, agents I and 2 eat object a, whereas agents 3 and
4 eat object b; at t = 1/2, unit amounts of a and b are consumed,
so a and b are no longer available; from this point on, agents I and
2 eat object c (their best available object) and agents 3 and 4 eat
object d. At t = 1, all of the objects are consumed, and each agent
has consumed a unit amount. The resulting random assignment is
exactly the one shown earlier. The PS mechanism always finds an
assignment that is ordinally efficient, which means that there is no
other dominating (in the sense of first-order stochastic dominance)
assignment matrix. Morever, it determines an envy-free allocation
for precisely the same reason that the CEEI solution does: here,
the eating speeds are identical across agents. All this is achieved,

unfortunately, at the expense of strategyproofness: the PS mecha-
nism is not strategyproof, but is weakly strategyproof in the sense
that by reporting false preferences no agent can find an allocation
that dominates his allocation under true preferences. Furthermore,
Bogomolnaia and Moulin [7] show that no mechanism that is ordi-
nally efficient and strategyproof can treat equals equally, which is a
very strong and somewhat disappointing impossibility result. On the
positive side, however, a number of recent papers have examined
"large markets" (markets with many copies of each object, say): the
general flavor of these results is that, in appropriately chosen large
markets, RP becomes ordinally efficient or that PS becomes strate-
gyproof [10, 21].
In contrast to the literature on deterministic mechanisms for
house allocation, there are very few axiomatic characterizations of
these mechanisms. In particular, it is believed that RP is the only
mechanism satisfying equal treatment of equals, Pareto efficiency,
and strategyproofness.

5 A Unified Model: Probabilistic Mechanisms
In recent work, Athanassoglou and Sethuraman [6] discuss the fol-
lowing model that subsumes all the house allocation models dis-
cussed so far. There are n agents and n objects, and agent i is
endowed with eij units of house j, with each -,, e [0, 1]. To keep
things simple, assume that each agent owns at most the equivalent of
a full object, and that at most one unit of any object is owned by the
agents, so that the endowment matrix is a doubly sub-stochastic ma-
trix. Each agent i has (ordinal) preferences over the set of houses ex-
pressed by the complete and transitive relation >i. While this pref-
erence relation allows for indifferences, again for simplicity, assume
a strict preference ordering for each agent. What we wish to find is
an allocation, which, as before, is described by an assignment matrix,
with the rows indexing the agents and columns indexing the objects;
like the endowment matrix, the assignment matrix will be a doubly
sub-stochastic matrix, as we assume that each agent is interested in
at most the equivalent of one object. Observe that this model gen-
eralizes the most prominent models studied in the house allocation
literature. In particular, if the endowment matrix is a permutation
matrix, we recover the Shapley-Scarf [35] housing market model; if
the endowment matrix is identically zero, we get the house alloca-
tion model; and if the endowment matrix is a sub-stochastic matrix
with entries in {0, 11, we get the the house allocation problem with
existing tenants, considered by Abdulkadiroglu and Sonmez [3] and
Yilmaz [39]. As before, the objective is to find compelling allocation
mechanisms and explore their properties such as efficiency, strat-
egyproofness, fairness. Furthermore, as some of the agents enter
the market with an endowment individual rationality an agent's fi-
nal allocation should (weakly) dominate his endowment becomes
important. Thus, the input to the mechanism is a strict preference
profile and an endowment matrix, where the preference orderings
are assumed to be private information, but the endowments are not.
A motivating example to keep in mind is the following situation: sup-
pose the final assignment of objects to agents will be made based on
a given fractional assignment matrix E, so that agent i will receive
object j with probability (,,. Interpreting this fractional assignment
matrix as the endowment of the agents, the mechanism computes an
alternative assignment matrix in which each agent's random alloca-
tion stochastically dominates her endowment, yielding a "superior"
lottery for each agent (this is the individual rationality requirement).
Ordinal efficiency of the proposed mechanism implies that this new
lottery cannot be improved upon for all the agents simultaneously.
As the agents come to the market with different endowments, in-
terpreting the "fairness" requirement is challenging. Consider the


following instance with three agents {1,2,3} and three objects
{a, b, c}. Agent I prefers a to b and b to c; agents 2 and 3 prefer
b to a and a to c. The initial endowments are specified in braces,
next to the preference ordering. Here, agent I is endowed with b,
agent 2 with a, and agent 3 with c.

1: a> b > c {b
2: b> a >c {a}
3: b> a >c {c}

It is clear that the only individually rational and efficient assignment
is one in which I gets a, 2 gets b and 3 gets c. Clearly agent 3 will
envy both agents I and 2. However, this envy is not justified because
it is not possible for agents I and 2 to give up any portion of their
endowments to agent 3, receive a positive share of house c and still
maintain individual rationality. In contrast, in the following example,

1: a> c >b {b}
2: b> c >a {a}
3: b> a >c {c}

the assignment discussed earlier giving a to I, b to 2, and c to 3
- is still individually rational and efficient. However there are other
individually rational and efficient allocations because agents I and 2
are willing to give up some of b and a respectively for any object in
the sets {a,c} and {b,c} respectively. In this context, if all of c is
allocated to agent 3, then this agent could justifiably envy agents I
and 2. This is because instead of giving agents I and 2 their best ob-
jects, the mechanism could have found a different allocation in which
agents I and 2 do a little worse, still maintain individual rationality,
and agent 3 does a little better. In particular, the assignment

a b c

1 1
2 0 1 1
S 2 2
3 0

is individually rational, efficient, and is envy-free.
Notice that in the TTC algorithm agents I and 2 would swap their
endowments, leaving agent 3 with object c. Thus the key difference
between the TTC method and the probablistic solution just obtained
is in the interpretation of the endowments: TTC allows the agents
to trade their objects directly, but this may not be possible always.
(As a concrete example in school assignment in the US: residing in
a certain neighborhood confers a right to attend a particular pub-
lic school, but this right is not tradeable.) In such environments the
only role of the initial endowment is that of a guarantee: each agent
is assured of a final assignment that is at least as good as his initial
endowment, but owning a "superior" object does not necessarily
imply a "superior" allocation.
For the house allocation problem with fractional endowments,
Athanassoglou and Sethuraman [6] design an algorithm to find an
assignment that is individually rational, ordinally efficient, and elimi-
nates justified envy of the sort discussed in the example above. (The
formal definition appears in Yilmaz [39].) This algorithm falls under
the general class of simultaneous eating algorithms (of which the
PS mechanism is a special case). In particular, it allows each agent
to "eat" her most preferred available object at rate I, as long as
there is some way to complete the assignment so that the individual ra-
tionality constraints are not violated; this continues until some object is
completely consumed, or some individual rationality constraint is in
danger of being violated. In the latter case the agents, whose contin-
ued consumption of their best available houses would violate some
individual rationality constraint, are forbidden from consuming their

most preferred houses even if they are available, and they move on
to their next best house. It is interesting that these events can be
tracked by solving a parametric flow problem on an appropriately
defined network, see [6] for a formal description. The implication is
that for this very general model, there is a mechanism that always
finds an ordinally efficient, individually rational allocation that avoids
justified envy. The price for this generality, however, is that strat-
egyproofness in any form is not possible. In fact, it is known that
even the bare basic requirements of efficiency and individual ratio-
nality are already incompatible with strategyproofness in this gen-
eral model. These and other impossibility results are also discussed
in [6].

6 Applications
The models discussed so far have a number of applications, the most
prominent ones being the assignment of schools to students [4] and
the organization of kidney exchanges [32]. There are a wealth of
papers and articles that describe these applications in greater de-
tail. We focus on two issues, one specific to each of these applica-

6.1 Kidney Exchange
Suppose there is a patient who needs a kidney, has a willing donor,
but the donor's kidney is not a medical match for the patient. Sup-
pose there is a second such patient-donor pair, and suppose that
the first donor can give a kidney to the second patient, and the first
patient can receive a kidney from the second donor. If such pairs
can be identified, then the two transplants can be performed, and
neither patient enters the waiting list for kidneys. This motivates the
need for an organized kidney exchange in which such patient-donor
pairs can make their presence known. The connections to the hous-
ing market problem is very clear: the patients are the agents, the
donors are the objects, and the quality of the match will serve as
the preference ranking of the agents. This simple connection already
raises a lot of questions: for example, what if one of the patients
has a willing donor who makes a donation, and in return, the patient
does not get a kidney but gets a preferred position in the wait-list.
Similarly, what if there is an altruistic donor who is not tied to any
patient? These considerations already call for a model in which some
agents may own objects, but others come to the market with no ob-
ject, and some objects are in the market unattached to any owner.
This is called the house allocation model with existing tenants and
was explored even before the kidney exchange application was for-
mally studied.
An important constraint in the kidney exchange problem is that
all the transplants involved in an exchange must be performed si-
multaneously. This, along with other practical considerations, makes
pairwise kidney exchanges more attractive. Furthermore, as a first
approximation to the actual problem, we may simply check whether
a particular kidney is a medical match for a particular patient or not.
This naturally leads to a matching problem in a graph, but with a
non-standard objective. In terms of the house allocation model, this
is an extreme special case in which each agent classifies each object
as either acceptable or unacceptable; any acceptable object is just as
good as every other acceptable object.
Matching models with this extreme form of indifference have
been studied by Bogomolnaia and Moulin [8] and Katta & Sethura-
man [20] for the case of a bipartite graph, in which agents represent
one side, and the objects the other. The work of Katta & Sethu-
raman [20] provides a complete solution to the house allocation
problem in which agent preferences have indifferences that are signif-

April 2010

icantly more general. In this article, we discuss only the special case
of extreme indifference, also called the case of dichotomous prefer-
ences. Each agent indicates only the subset of objects that she finds
acceptable, each of which gives her unit utility; the other objects are
unacceptable, yielding zero utility. Any solution is simply evaluated
by the expected utility it gives to each agent, which is simply the
probability that she is assigned an acceptable object. Suppose the
preference profile in a given instance is such that some set of three
agents have only two acceptable objects among them. Then, it is ob-
vious that the combined utilities of these three agents cannot exceed
2. The general algorithm for solving the problem builds on this trivial
observation: it consists of locating such a bottleneck subset of agents,
and allocating their acceptable objects amongst them in a fair way,
eliminating these agents and their acceptable objects, and recursively
applying the idea on the remaining set of agents and objects. It is an
elementary exercise to show that these problems can be solved as
the problem of finding a maximum flow in the following network:
there is a node for each agent and for each object, and there is an
infinite-capacity arc from agent node i to object node j if i finds j
acceptable. Augment the network by adding a source node s, a sink
node t; arcs with capacity A > 0 going from s to each agent node,
and arcs with unit capacity going from each object node to the sink
t. We view A > 0 as a parameter, and study the minimum-capacity
s t cuts (or simply minimum cuts) in the network as A is varied.
Let A* be the (smallest) breakpoint of the min-cut capacity function
of the parametric network. Clearly, A* < 1. If A* = 1 then every
agent can be assigned an acceptable object; otherwise, the agents
on the source-side of the min-cut form a bottleneck set, and each of
them can only be matched to an acceptable object with probability
A*. The objects they get matched to with these probabilities is given
by any flow with value nA* in this network. Eliminating these agents
and they objects they find acceptable (every one of those objects
will necessarily be removed from the problem), we get a reduced
problem on which we apply the same algorithm. The breakpoints
of the min-cut capacity function of a parametric network are well-
understood [13]. Efficient algorithms to compute these breakpoints
have been discovered by several researchers, see for example Ahuja,
Magnanti, and Orlin [5] and the original paper of Gallo, Grigoriadis
and Tarjan [I 3].
This mechanism just described is coalitionally strategyproof, envy-
free, and efficient in a very strong sense: this mechanism finds a util-
ity vector for the agents that Lorenz dominates the utility vector
found by any other mechanism. (A vector x Lorenz dominates a
vector y if the sum of the k smallest components of x is (weakly)
larger than the sum of the k smallest components of y, for every
k > 1.) In fact, the house allocation problem with dichotomous pref-
erences is closely related to the sharing problem, first introduced
by Brown [9]. Brown's work was partly motivated by a coal-strike
problem: during a coal-strike, some "non-union" mines could still
be producing. In this case, how should the limited supply of coal
be distributed equitably among the power companies that need it?
Since power companies vary in size, it would not be desirable to
give each power company the same amount of coal. Moreover, even
if such an equal sharing was desirable, the distribution system may
not allow for a perfect distribution because of capacity constraints.
Brown [9] modeled the distribution system by a network in which
there are multiple sources (representing the coal-producers), mul-
tiple sinks (representing power companies), and multiple transship-
ment nodes; each edge in the network has a capacity which is an
upper bound on the amount of coal that can traverse that edge.
Also, each sink node has a positive "weight" reflecting its relative
importance, and the utility of any sink node is the amount of coal
it receives divided by its weight. The goal is to distribute the sup-

ply of coal so as to maximize the utility of the sink that is worst-
off. Megiddo [23, 24] considered the lexicographic sharing problem,
where the objective is to lexicographically maximize the utility vec-
tor of all the sinks, where the kth component of the utility vector
is the kth smallest utility. The algorithm described earlier is pre-
cisely the one that finds a lex-optimal flow for this sharing prob-
In the case of kidney exchanges, the associated graph is really non-
bipartite, and this has been formally worked out by Roth, Sonmez
and Unver [33] using classical combinatorial tools. The ideas de-
scribed for the bipartite case, however, essentially carry over, yield-
ing a simpler proof of many of those results.

6.2 School Choice
Abdulkadiroglu and Sonmez [4] considered the problem of assigning
students to schools and formulated it as a matching problem, and
advocated two broad types of solutions for it. One is the familiar and
well-studied stable matching model, introduced earlier by Gale and
Shapley [12]. The other is the TTC algorithm with object priorities
reflected by an inheritance table. In both cases, the students were
the agents whose preferences had to be elicited. The schools were
viewed as passive objects whose priority lists were exogenous. After
an extensive evaluation of both mechanisms, the public school sys-
tem in NYC (and in some other American cities) has been using the
Gale-Shapley mechanism for assigning students to high schools [I].
There is, however, a supplementary round that is meant to assign
the applicants who are unassigned after the main round. The assign-
ment process in the main round takes into account several factors
including the student priorities at each school based on standard-
ized test scores, and is therefore a two-sided matching problem.
In contrast, the assignment process in the supplementary round is
quite simple: all unassigned applicants are invited to rank order high
schools with vacant capacity; all students at this stage have the same
priority to attend any school. Thus we are led naturally to a setting
in which agents rank heterogeneous "objects," for which they have
equal claims. Motivated by these considerations, Pathak [28] recently
studied two "lottery" mechanisms: first is the single lottery mecha-
nism, in which a single random ordering of the agents is drawn; any
ties (at any school) are broken in favor of the student whose lottery
number is lower. A natural alternative which seems fairer to the
students at first glance is the multiple lottery mechanism, which al-
lows each school to conduct its own lottery. The actual assignment
is made by the TTC mechanism applied to the preference profiles of
the students and the priority profiles of the schools [4]. Pathak [28,
pp. 3] notes that during the course of the design of the new assign-
ment mechanism, policymakers from the Department of Education
believed that the single lottery mechanism is less equitable than the
multiple lottery mechanism. Remarkably, Pathak shows that, for the
special case of the problem in which each school has exactly one
vacant spot, the distribution of assignments is exactly the same un-
der both mechanisms! In a recent paper, Sethuraman [36] proves
the equivalence of the single and multiple lottery mechanisms in
full generality. This result is proved by introducing a new class of
mechanisms called Partitioned Random Priority (PRP). Under the PRP
mechanism, we are given an arbitrary partition Si,S2,...,Sk of the
"schools;" the schools within each Si use a common lottery, and dis-
tinct Si's use an independent lottery. (Note that if each school is in
a partition by itself we recover the multiple lottery mechanism; if all
the schools belong to a single partition, we recover the single lottery
mechanism.) The key result is that the distribution of assignments
under the PRP mechanism is the same, regardless of the partition of
the schools. The analog of Pathak's result when schools have multiple
seats follows: if there are k schools and school i can admit qi stu-


dents, then make qi copies of school i, and let Si consists of these
qi copies.

7 Conclusion
The literature on house allocation problems is extensive, neverthe-
less a number of challenging open questions remain. One important
direction for future research is the study of dynamic versions of
these problems. For example, the kidney exchange problem is really
a dynamic problem, in which donor-patient pairs arrive or leave, and
patients need to make a trade-off between accepting a less-preferred
option now versus waiting for a potentially better match in the fu-
ture. While there are a few recent papers that take up such prob-
lems [40], much remains to be done. Another intriguing question
is to obtain an effective characterization of the domains for which
the Gibbard-Satterthwaite impossibility result can be avoided. The
house allocation and related problems represent one such general
class of problems, but there are many others. The work on integer
programming approaches to social choice [37] is a first step in this
direction, but we are still far from a complete understanding of this

This research was supported by the National Science Foundation
under grant CMMI-0916453.

Jay Sethuraman, IEOR Department, Columbia University, New York, NY;

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Discussion Column

Garud lyengar and Anuj Kumar

Parametric Network Flows in Adword


I Introduction
Sponsored search advertising is a major source of revenue for in-
ternet search engines. Close to 98% of Google's total revenue of
$6 billion for the year 2005 came from sponsored search ads. It is
believed that more than 50% of Yahoo!'s revenue of $5.26 billion
was from sponsored search advertisement. Sponsored search ads
work as follows. A user queries a certain adword, i.e., a keyword

April 2010

relevant for advertisement, on an online search engine. The search
engine returns the links to the most "relevant" webpages and, in ad-
dition, displays certain number of relevant sponsored links in certain
fixed "slots" on the result page. Every time the user clicks on any of
these sponsored links, she is taken to the website of the advertiser
sponsoring the link and the search engine receives certain price per
click from the advertiser. The likelihood that a user clicks an ad is
a function of the slot; therefore, advertisers have a preference over
which slot carries their link. The click likelihood is also a function
of the exogenous brand values of the advertisers; therefore, search
engines prefer allocating more desirable slots to advertisers with
higher exogenous brand value. Thus, search engines need a mech-
anism for allocating slots to advertisers. Since auctions are very ef-
fective mechanisms for revenue generation and efficient allocation,
they have become the mechanism of choice for assigning sponsored
links to advertising slots.
Adwords auctions are dynamic in nature the advertisers are al-
lowed to change their bids quite frequently. In this note, we design
and analyze static models for adword auctions. We use the dominant
strategy solution concept in order to ensure that the static model
adequately approximates dynamic adword auctions. We consider the
case where the private known valuation for a click is independent of
the ad slot, i.e., the advertiser values clicks but is indifferent about
where that click originated. The search engine wants to assign slots
to advertisers using an auction that induces the advertisers to be
truthful, i.e., the auction is incentive compatible, and leaves them no-
worse off than if they had not participated in the adword auction, i.e.,
it is rational for advertiser to participate in the auction. Among all
these auctions, the auctioneer wants to choose one that maximizes
revenue. In this note we show that the revenue maximizing incen-
tive compatible, individually rational, auction can be implemented in
a computationally tractable manner using parametric network flows.

2 Revenue Maximizing Adword Auction
Suppose there are n advertisers bidding for m(< n) slots on a
specific adword. Let ci denote the click-through-rate when adver-
tiser i is assigned to slot j. For convenience, we will set ci,m+ = 0
for all i = 1,...,n. We assume that for all bidders i, the rate cy
is strictly positive and non-increasing in j, i.e., all bidders rank the
slots in the same order. The rates ci, for all (i, j) pairs i = 1..., n,
j 1,...,m, are known to the auctioneer, and only the rates
(CilCi2,...,Cim) are known to bidder i, i.e., each bidder only
knows her click-through-rates.
We assume that the true per-click-value vi of advertiser i is pri-
vate information and is independent of the slot j. We assume an in-
dependent private values (IPV) setting with a commonly known prior
distribution function that is continuously differentiable with density
f(vi,...,) = n i fi(vi) : n ++. Even through we use
dominant strategy as the solution concept, we still need the prior
distribution in order to select the optimal revenue maximizing mech-
We restrict attention to direct mechanisms the revelation prin-
ciple guarantees that this does not introduce any loss of generality.
In this setting, advertiser i submits a bid bi which is the amount she
is willing to pay for a ad-slot. Let b e Rn denote the bids of the
n bidders. An auction mechanism for this problem consists of the
following two components:
I. An allocation rule X : Rn {0, 1}nxm such that ina Xy(b)
1, for all j 1,...,m, and na L Xij(b) < 1, i 1,...,n. Thus,
X(b) is a matching that matches bidders to slots as a function
of the bid b. We denote the set of all possible matching of n
advertisers to m slots by Mnm.

2. A payment function T : Rn In that specifies the amount each
of the n bidders pays the auctioneer. Note that this payment
function depends on the bids of all the advertisers.
In Lemma I we show that the payment of a bidder who is not al-
located any slot can be set to zero without any loss of general-
ity. Thus, we can define the per click payment ti of advertiser i as
ti (b) T(b) For all i 1,...,n, vi e B and b i c Rn-1

ui(b,v;(X, T),b i) (cijv -ti(b,b i))Xiy(b,b i) (I)

denote the utility the advertiser i of type v who bids b and all
the other advertisers bid b i. We restrict attention to mechanisms
(X, T) that satisfy the following two properties:
I. Incentive compatibility (IC): For all i = 1,...,n, vi e R+, and
b i e R 1, vi e argmaxbr, {ui(b, v; (X,T),b i)}, i.e., truth
telling is ex-post dominant.
2. Individual rationality (IR): For all i = 1,...,n, vi e IR+, and
b i c Rn-1 maxbcR [ui(b,v;(X,T),b i)} > 0, i.e., we implic-
itly assume that the outside alternative is worth zero.

vtjv p

al a2 a3

Figure I. Incentive Compatibility: Allocated click-though-rate as a function of

Lemma I. The following are equivalent characterizations of IC alloca-
tion rules.
(a) The click-through-rate I7l cijXij (vi, v ) is non-decreasing in vi
for all fixed v i.
(b) For all i and vi there exist thresholds ai,m+l 0 < aim(v i) <
ai,m 1 (v i) < < ai, (vi ) < oo such that bidder i is assigned
slot j iffyi e i, (v i),,i, i+ (v i)].
(c) For each advertiser i, there exist slot prices {pij(vi)}n I of the form

k m
Pj(v-i) c- Y (aik(V-i) ai,k+i(V-i))(cij -Ci,k+1)

S (Cik C- i,k 1 +,i, ,( i) (2)
k -j
cif k=j

where 0 < aim(vi) < ai,m- (Vi) < .. < ai,1 (vi) < oo such
that advertiser i self-selects her assigned slot.

Note that the price pij 0 for j m + 1, i.e., the advertiser
does not pay anything if she is not assigned a slot. Lemma I gives
us a method for constructing incentive compatible auctions first
select an allocation rule that guarantees I cijXij (vi, vi ) is non-
decreasing in vi for fixed v i; next, compute the i,, ((vi) and


charge the slot prices ij (v i). The proof of Lemma I follows from
Holmstrom's Lemma (see, p.70 in [3]). For details see [2].
Our next task is to select a revenue maximizing auction. From [4],
it follows that expected revenue of the auctioneer under any domi-
nant strategy incentive compatible allocation rule X is given by

[n mj
E co

IF(vi) v) + u,(0.v ,)

where fi: R+ R++ is the prior density of vi, i 1,..., n. Let

X* (v) e argmax Ci vi
XCMWn i[ij V

S Fi(vi) (3)
fi (i) rT (3)

Suppose the virtual valuations per click vi(vi) vi (v,)
are non-decreasing. Then the allocation rule X* results in a non-
decreasing total-click-through for each of the bidders. Hence,
part (a) in Lemma I implies that X* is IC. Since the point-
wise maximum is an upper bound on any expected revenue
maximizing allocation, (X*,T*) is expected revenue maximizing,
dominant strategy incentive compatible, individually rational al-
location rule with per-click prices given by (2). When the vir-
tual valuations v(vi) are non-monotonic, the revenue maximiz-
ing mechanism can be constructed by first ironing (see [4])
the virtual valuation to obtain a non-decreasing virtual valua-
tions Vi(vi) and then solving (3). Given v, (3) can be solved,
or equivalently X*(v) can be computed, in O(m2n) opera-
tions as a solution to an optimal weighted matching prob-
Next, we show how to efficiently compute the thresholds
1., (vi) and the payments Pij(vi) corresponding to the rule X*.
Recall that ,i, (v i) denotes the threshold value that ensures that
advertiser i will be assigned to a slot j or better. Fix an advertiser
io. Consider the parametric assignment problem in A:

I m n m 1
max A CjXio+ k(Vk)CkjXkj (4)
Xc m [ j=1 k=l,k-ij=l

This is an LP and the solution Xioj is a piece-wise constant in A. Let
{Aj} 1 denote values of A such that Xioj = 1 for all A e (Aj, Aj+ i].
Since Vio is non-decreasing, the thresholds i, i = v9i(Aj). How-
ever, it is not immediately obvious that the thresholds {Aj} can be
efficiently computed.
The assignment problem (4) can be formulated as a parametric
minimum cost network flow problem on an appropriately defined
graph. Using standard results for parametric network flows, (see
Exercise I 1.48 in [I] and [2] for details) one can show that there
exists an algorithm OPTMATCH that takes as input (io,v i0,c) and
computes the thresholds in O(m2n) operations. Once we have
the thresholds, computing the slot prices is easy. Algorithm COM-
PUTEPRICES cycles through each of the advertisers, calls OPT-
MATCH to get the thresholds, and then uses (2) to compute the
prices. Thus, Algorithm COMPUTEPRICES computes the slot prices
in ((m2n2) operations. Thus, we have established the goal that we
set ourselves in this note -we have an efficient algorithm for running
a revenue maximizing adword auction!

I: z (i'(vi),...., n(vn)), Ci,m+ 0, ai,m+l 0 Vi.
2: for i = 1 to n do
3: a OPTMATCH(i,z i,c).
4: for j = 1 to m do
5: aij Jil(i(j)
6: if aijy oo then
7: aij minj 8: end if
9: end for
10: for j = 1 to m do
II: Pij -- 2 j (ai ai,k+ 1)(i ci,k+1)
12: end for
13: end for

Garud lyengar, Industrial Engineering and Operations Research Department,
Columbia University, New York, NY-10027, USA. gi l@columbia.edu
Anuj Kumar, Barclays Capital, New York, NY-10027, USA.

[I] Ahuja, R. K., Magnanti, T., and Orlin, J. (1993). Network Flows: Theory, Algo-
rithms and Applications. Prentice Hall, NJ.
[2] lyengar, G. and Kumar, A. (2006). Characterizing optimal adword auctions.
CoRR abs/cs/061 1063.
[3] Milgrom, P R. (2004). Putting Auction Theory to Work. Cambridge University
Press, Cambridge, UK.
[4] Myerson, R. B. (1981). Optimal auction design. Mathematics ofOperations
Research, 6(1):58-73.

Dimitri P Bertsekas and Zhi-Quan (Tom) Luo

Paul Tseng, 1959-2009

Paul Tseng, Professor of Mathematics at the University of Washing-
ton, Seattle, has been missing since August I3, 2009 while kayaking
in the Yantze river near Lijiang, in Yunnan province of China. His
friends were alerted when he did not show up for an invited talk
on August 17 at an International Conference on Numeric Optimiza-
tion and Numeric Linear Algebra in Lijiang, hosted by the Chinese
Academy of Sciences. The sad news quickly propagated through the
optimization community, where Paul is greatly loved by his many
friends and widely respected for his research accomplishments. His
scheduled semi-plenary lecture at the ISMP meeting in Chicago (im-
mediately after the Lijiang conference) had to be canceled.
Following an intensive search, helped by his close friend and col-
laborator, Tom Luo of the University of Minnesota, the events lead-
ing to Paul's disappearance were pieced together: Paul flew from
Seattle on August I I, through Shanghai and Chengdu, into Lijiang
on August 13. Then from the Lijiang airport he took a taxi directly
to a remote location, near the Jin'an Bridge on the Jinsha river (a
tributary of the Yantze river), where he launched his kayak in rapid
waters at about 4PM on August 13. He had planned to kayak for
three days, or about 400 kilometers, on the Jinsha river through a
beautiful but mostly uninhabited mountainous region. Paul, an avid
outdoorsman and very seasoned kayaker, appeared to struggle with
the water from the beginning (based on eye-witness accounts), and
then disappeared from view. His kayak and backpack were found on
August 30 a few kilometers downstream from where he entered. It
is believed that Paul was a victim of an unfortunate accident. He is
survived by his mother and his sister Nora.

April 2010

Paul grew up in Taiwan and Canada, and worked primarily in the
United States. His family came from China to Taiwan, where he was
born on Sept 21, 1959 in Hsin-Chu. Later his family moved to Taipei,
where Paul went to elementary school. Paul and his family moved to
Vancouver, Canada in December 1970, where Paul graduated from
high school in 1977.
Paul was well known for his adventurous and unconventional trav-
els, often using bicycle and kayak. In the years 1986-2008, he took
long bicycle trips through Europe, Central America, and Kenya, and
kayaking trips in the Danube, the Mekong, the Baltic Sea, the Nile,
the Red Sea, Vancouver Island, the Yellow River in China, and the
Rio Madre de Dios (a headwater tributary of the Amazon River
in Peru). He kayaked for long distances (as examples, from Laos
to the Mekong delta in Vietnam, and from Prague to the Danube
delta in Romania), often mixing with local people on the way and
sharing their lifestyles. He brought back many pictures and stories,
which can be found at his website http://www.math.washington.edu/
-tseng/personal.html. His ambition was to kayak in all the major
rivers of the world.
Paul had several other nonprofessional interests. He liked sports
and he was well-known for his expert tennis game. He also had
a strong interest in music, particularly in playing the piano, and he
liked drawing, painting, pottery making and woodcarving. He spent a
few summers drawing portraits in Stanley Park in Vancouver, and he
had a summer job making wood carvings of West Coast animals. He
was a minimalist in life, with a deeply held commitment to environ-
mentalism and noble causes (he had frequently "walked for hunger"),
and he tended his garden and beautiful roses with great care.
Paul received his B.Sc. from Queen's University (Kingston, On-
tario) in Mathematics in 1981, and his Ph.D. from the Operations Re-
search Center of the Massachusetts Institute of Technology (Cam-
bridge, MA) in 1986. After working for one year at the University of
British Columbia, he spent three years at the Massachusetts Institute
of Technology as a postdoc in the group of Dimitri Bertsekas and
John Tsitsiklis, working on optimization and distributed computa-
tion. Paul moved in 1990 to the University of Washington's Depart-
ment of Mathematics, where he worked alongside Terry Rockafellar
and Victor Klee.
Paul's research has been mainly in continuous optimization, with
side interests in discrete optimization, distributed computation, and
network and graph algorithms. He is widely recognized by his peers
as one of the foremost optimization researchers of his generation,
at a time of great progress in his field. He has published extensively
(over 120 journal papers), and his research subjects include among
o Efficient algorithms for structured convex programs and network
flow problems,
o Complexity analysis of interior point methods for linear program-
o Parallel and distributed computing,
o Error bounds and convergence analysis of iterative algorithms for
optimization problems and variational inequalities,
o Interior point methods and semidefinite relaxations for hard
quadratic and matrix optimization problems, and
o Applications of large scale optimization techniques in signal pro-
cessing and machine learning.
Paul's Ph.D. thesis was on network optimization methods and
related monotropic programming problems. He coauthored with
his advisor Dimitri Bertsekas, a series of papers on relaxation meth-
ods and monotropic programming, as well as a publicly available
network optimization program, called RELAX, which is widely used
in industry and academia for research purposes. Among his other
research accomplishments, Paul, together with Tom Luo, resolved

4 Springer
the language of science

An Interactive Course
in Combinatoral



An Interactive Course in
Combinatorial Optimization

CATBox consists of a software system
for animating graph algorithms and a
course book developed simultaneously.
The software system presents both the
algorithm and the graph. The course
book introduces the background

necessary for understanding the algorithms.

1st Edition., 2010. XII, 190 p., Softcover
ISBN 978-3-540-14887-6 0 $59.95

Visit http://schliep.org/CATBox for more information

Easy Ways to Order forth Americas Write: Springer Order Department, PO Box
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> Prices are subject to change without notice. All prices are net prices.

a long-standing open question on the convergence of matrix split-
ting algorithms for linear complementarity problems and affine vari-
ational inequalities, and was the first to establish the convergence of
the affine scaling algorithm for linear programming in the presence
of degeneracy. Furthermore, in a series of papers with Tom Luo, he
developed a theory of error bounds and used it creatively to yield a
strong convergence rate analysis for a broad class of iterative algo-
rithms including the proximal splitting methods and the successive
projection methods to convex sets, both of which find contempo-
rary applications in compressive sensing and image processing. He
was widely admired for his creative work and his productivity, and
was well-liked for his cheerful and friendly manner. He has had close
collaborations with several colleagues, and he served the community
as a conscientious and hard-working editor in several top optimiza-
tion journals for many years.
A special workshop called "Large-scale optimization: Analysis,
algorithms and applications" is planned in Paul's honor for May
21, 2010, at Fudan University, in Shanghai, China, where sev-
eral of his collaborators will participate and present research re-
lated to topics where Paul's work has had a major impact. See
http://www.se.cuhk.edu.hk/Workshop2010/home.html where you
can view and upload photos of Paul, and leave messages. You
may also visit http://www-optima.amp.i.kyoto-u.ac.jp/ORB/issue34/
issue34.html where you can read messages from Paul's friends and
colleagues, and have the opportunity to leave your own message.

Dimitri P. Bertsekas, McAfee Professor of Engineering, MIT, 77 Mas-
sachusetts Ave., Cambridge, MA 02139. dimitrib@mit.edu
Zhi-Quan (Tom) Luo, Department of Electrical and Computer Engineer-
ing, University of Minnesota, 200 Union ST SE, Minneapolis,MN 55455.


M. JungerT Liebling, D. Naddef, G. Nemhauser,
W. Pulleyblank, G. Reinelt, G. Rinaldi, L.Wolsey (Editors)
50 Years of Integer Programming 1958-2008

In 1958, Ralph E. Gomory transformed the field of integer programming when he published
a short paper that described his cuttingplane algorithm for pure integer programs and
announced that the method could be refined to give a finite algorithm for integer program
ming In January of 2008, to commemorate the anniversary of Gomory' seminal paper, a
special session celebrating fifty years of integer programming was held in Aussois, France,
as part of the 12th Combinatorial Optim ization Workshop. This book is based on the material
presented during this session.
50Years of Integer Program ming offers an account of featured talks at the 2008 Aussols
workshop, namely
* Michele Conforti, Gerard Cornuejolsand Giacomo Zam bell: Polyhedral Approaches to
Mixed Integer Linear Programming
* William Cook 50+ Years of Combinatorial Integer Programming
* Franois Vanderbeckand Laurence A. Wolsey: Reformulation and Decomposition of
Integer Programs
It includes a DVD containing a recording of the three original lectures as well as a panel
discussion with six pioneers.
The book contain reprints of key historical articles together with new introductions
and historical perspectives by the authors: Egon Balas, Michel Balinski Jack Edmonds,
Ralph E. Gomory Arthur M. Geoffnon, Alan J. Hoffman &Joseph B. Kruskal, Richard M. Karp,
HaroldW. Kuhnand Ailsa H. Land & Alison G. Dog.
It also contains written versions of survey lectures on six of the hottest topics in the field
by distinguished members of the IP community:
SFriednch Eisenbrand: Integer Programming and Algorithmic Geometry of Numbers
* Raymond Hemmecke Matthias Koppe Jon Lee, and Robert Weismantel Nonlinear
Integer Programming
* Andrea Lodi: Mixed Integer Programming Computation
* Franols Margot: Symmetry in Integer Linear Programming
SFranz Rendl: Semidefinite Relaxations for Integer Programming
* Jean Philippe P Richard and Santanu S. Dey: The Group-Theoretic Approach to Mixed
Integer Programming
Integer programming holds great promise for the future and continues to build on its
foundations. Indeed, Gomory's finite cuttingplane method for the pure integer case is
currently being reexamined and is showing new promise as a practical computational
m ethod. This books a uniquely useful celebration of the past present and future of this
important and active field. Ideal for students and researders in mathematics computer
science and operations research, it exposes mathematical optimization, in particular integer
programming and combinatorial optimization, to a broad audience

ISBN 978 3 540 68274 5

9783540 682745
) springer.com

- TI
-. C_

L_ D. e i Pa.d

50Years of

Inteaer Pro0ramming

0o (D






No-Charge Access to IBM ILOG

Optimization Products for Academics

IBM now provides full-version IBM ILOG Optimization products via
IBM's Academic Initiative program, which provides no-charge access
to full-version software and professionally developed courseware for
non-commercial research and teaching purposes.
Registered members of the IBM Academic Initiative can access
and search for IBM ILOG Optimization products in the Software
Catalog: https://www.ibm.com/developerworks/university/software/
Those who are not current members, should first register
and become an Academic Initiative member: http://www.ibm.com/
To learn how to install the software and license keys for IBM ILOG
Optimization products, Academic Initiative members should follow
the Step by Step Process and Quick Start Guide: https://www.ibm.
For further questions, please contact IBM directly at aioptim@us.

MIPLIB20 10 Call for contribution

Since its first release in 1992, the MIPLIB has become a standard test
set used to compare the performance of mixed integer linear opti-

.I 1958-2008



mization software and to evaluate the computational performance
of newly developed algorithms and solution techniques.
Seven years have passed since the last update in 2003. Again, the
progress in state-of-the-art optimizers, and improvements in com-
puting machinery have made several instances too easy to be of fur-
ther interest. New challenges need to be considered!
Last year a group of interested parties including participants from
ASU, COIN, FICO, Gurobi, IBM, and MOSEK met at ZIB to discuss
the guidelines for the 2010 release of the MIPLIB. It will be the fifth
edition of the Mixed-Integer Programming LIBrary.
Therefore, we are looking for interesting and challenging (mixed-)
integer linear problems from all fields of Operations Research and
Combinatorial Optimization, ideally ones which have been built to
model real life problems. We would be very happy if you contributed
to this library by sending us hard and/or real life instances.
We have recently opened our submission web page and are look-
ing forward to your contributions: http://miplib.zib.de/miplib2010

24th European Conference on

Operational Research (EURO XXIV)

Lisbon, Portugal, July 11-14, 2010. The 24th European Conference
on Operational Research (EURO XXIV) is organized by EURO (The
Association of European OR Societies) and APDIO (The Portuguese
OR Society), with the support of the Faculty of Sciences of the Uni-
versity of Lisbon and CIO (Operational Research Centre, Portugal).

April 2010

The Programme Committee (chaired by Silvano Martello) and the
Organizing Committee (chaired by Jose Paixao), are preparing a high
quality scientific programme and an exciting social programme for
the Conference.
Plenary Speakers: Harold W. Kuhn and John F Nash, Jr.
Invited Speakers: Fran Ackermann, Noga Alon, James Cochran,
Elena Fernandez, Pierre Hansen, Martine Labbe, Nelson Maculan,
Michel Minoux, Arkadi Nemirovski, Stefan Reichelstein, Alexander
Shapiro, and Berthold Vockin.

Contacts: prog@euro20101isbon.org (Programme)
registration@euro20 Olisbon.org
(Registration, travel and accommodation)
info@euro20101isbon.org (General enquiries)
Website: www.euro20 I Olisbon.org

ICCOPT is a forum for researchers and practitioners interested in
continuous optimization, which takes place every three years. The
first version was held in 2004 at Rensselaer Polytechnic Institute
(Troy, NY, USA), while the second version was organized in 2007 at
McMaster University (Hamilton, Ontario, Canada).
The February 27 earthquake in Chile has not affected the con-
ference facilities, and the international airport in Santiago and other
transportation systems are operating normally. (Most damage oc-
curred 200-600 km south of Santiago.) Hence, ICCOPT 2010 will
go ahead as planned. Several deadlines have been extended (see be-
The Conference will feature a series of invited lectures, con-
tributed talks, and streams on specific subjects.
Plenary speakers include:
- Xiaojun Chen (The Hong Kong Polytechnic University)
- Roberto Cominetti (Universidad de Chile)
- Ignacio Grossman (Carnegie Mellon)
- Rene Henrion (Weierstrass Institute for Applied Analysis and
- Marco Locatelli (Universita di Parma)
- Zhi-Quan (Tom) Luo (University of Minnesota)
- Jorge Nocedal (Northwestern University)
- Mikhail Solodov (Instituto de Matematica Pura e Aplicada)

52th Workshop: Nonlinear Optimization,

Variational Inequalities and Equilibrium


July 2-10, 2010, Ettore Majorana Centre for Scientific Culture Interna-
tional School of Mathematics "G. Stampacchia" Erice, Italy. The Work-
shop aims to review and discuss recent advances in the development
of analytical and computational tools for Nonlinear Optimization,
Variational Inequalities and Equilibrium Problems, and to provide a
forum for fruitful interactions in strictly related fields of research.
Topics include constrained and unconstrained nonlinear optimiza-
tion, global optimization, derivative-free methods, nonsmooth opti-
mization, nonlinear complementarity problems, variational inequali-
ties, equilibrium problems, game theory, bilevel optimization, neural
networks and support vector machines training, applications in en-
gineering, economics, biology and other sciences.
The Workshop will include keynote lectures (I hour) and con-
tributed lectures (30 min.). Members of the international scientific

LT) 2010
Santlago, Chile
inter School

- Philippe Toint (Facult6s Universitaires Notre Dame de la Paix)
- Stefan Ulbrich (Technische Universitat Darmstadt)
- Luis Nunes Vicente (University of Coimbra)
A School on Continuous Optimization and Mathematical Model-
ing, addressed to PhD students and young researchers, will precede
ICCOPT-2010, on July 24-25. This School will provide an introduc-
tory but up-to-date perspective in two areas, namely,
- Optimization under uncertainty (Ruszczynski, Dentcheva,
Shapiro, Wets)
- Optimization in natural resources management (Alvarez, Amaya,
Ramirez, Gajardo, Rapaport)
Free registration will be provided for all School participants, and
they may also apply for free accommodation during the School and

Important Dates: Streams Submission: March 3 2010
Conference: Early registration: May 28, 2010
Abstract submission: May 10, 2010
School: Applications: April 30, 2010

More Information: For further information, including abstract and
stream submission, registration procedures, fees and accommoda-
tion sites, please see the conference web site at www.iccopt2010.
cmm.uchile.cl, or send email to iccopt2010@dim.uchile.cl.


community are invited to contribute a lecture describing their cur-
rent research and applications. Acceptance will be decided by the
Advisory Committee of the School.
Invited lecturers who have confirmed the participation are: Ernes-
to G. Birgin, Francisco Facchinei, Christodoulos A. Floudas, David
Gao, Diethard Klatte, Eva K. Lee, Marco Locatelli, Jacqueline Mor-
gan, Evgeni A. Nurminski, Jong-Shi Pang, Mike J. D. Powell, Franz
Rendl, Nikolaos V. Sahinidis, Katya Scheinberg, Marco Sciandrone,
Valeria Simoncini, Henry Wolkowicz, Ya-xiang Yuan.
A special issue of Computational Optimization and Applications
will be dedicated to the Workshop, including a selection of invited
and contributed lectures.
The scientific and organizing committee: Gianni Di Pillo (SAPIENZA
- University' di Roma, Italy), Franco Giannessi (Universita' di Pisa,
Italy), Massimo Roma (SAPIENZA Universita' di Roma, Italy).
Further information: www.dis.uniroma I.it/-erice2010,
erice20 10@dis.uniroma I .it
Poster of the Workshop:
www.dis.uniroma I .it/-erice20I0/poster.pdf

Conference on Computational

Management Science CMS20 10

July 28th-30th 2010, University of Vienna, Austria. The CMS con-
ference is an annual meeting associated with the journal of

Computational Management Science www.springer.com/business/
operations+research/journal/10287 published by Springer.
The aim of this conference is to provide a forum for theoreticians
and practitioners from academia and industry to exchange knowl-
edge, ideas and results in a broad range of topics relevant to the
theory and practice of computational methods, models and empiri-
cal analysis for decision making in economics, finance, management,
and engineering.
The CMS Best Student Paper Prize will be awarded at the CMS
conference. The prize is 300 EUR and the possibility of publication
in the journal of Computational Management Science. Papers can be
nominated by the supervisors of the students. Submission deadline
is June I st 2010. Only registered participants' papers will be consid-
ered for the prize.
Please visit www.univie.ac.at/cms2010/ for more details, abstract
submission, registration and accommodation.

SIAG/Optimization Prize:

Call for Nominations

The SIAM Activity Group on Optimization Prize (SIAG/OPT Prize)
will be awarded at the SIAM Conference on Optimization to be held
May 15-19, 201 in Darmstadt, Germany.
The SIAG/OPT Prize, established in 1992, is awarded to the au-
thor(s) of the most outstanding paper, as determined by the prize

Mixed Integer Programming 2010

July 26-29, 2010

Georgia Institute of Technology, Atlanta, Georgia


The 2010 Mixed Integer Programming
workshop will be the 7th in a series of annual CO
workshops held in North America designed to CO
bring the integer programming community Karer
together to discuss very recent developments
in the field. The workshop series consists of a Tobia:
single track of invited talks. Egon
We encourage anyone with interests in mixed Pierre
integer programming to participate. Space is
limited, and thus early registration is Miche
There will be a contributed poster session for
which we invite all participants to submit an Da
abstract; see the website for instructions. Matte
Space for posters is limited, so we may not be
able to accommodate all posters. There will Antor
be ample time for discussion and interaction Zong
between the participants during the
workshop. Ellis J
Thanks to the generous support by our Simg
sponsors, registration is free, and travel Jon L
support is available. Funding priority will be
given to students and postdocs who have
submitted poster abstracts.

,firmed speakers

s Achterberg
ele Conforti
eb Dash
l Espinoza
o Fischetti
iio Frangioni
hao Gu
e Kucukyavuz

Quentin Louveaux
Jim Luedtke
Andrew Miller
Michele Monaci
James Ostrowski
Marc Pfetsch
Jean-Philippe Richard
Martin Savelsbergh
Stefano Smriglio
Andrea Tramontani
Santosh Vempala
Juan-Pablo Vielma
Giacomo Zambelli

Important Dates
March 31
Deadline for poster
abstracts and requests for
travel support
April 30
Notification on poster
acceptance and travel
July 26-29

Organizing committee
Shabbir Ahmed
(Georgia Tech)
Ismael Regis de Farias Jr.
(Texas Tech University)
Ricardo Fukasawa
(University of Waterloo)
Matthias Koppe
(University of California, Davis)
Andrea Lodi
(University of Bologna)

April 2010

committee, on a topic in optimization published in English in a
peer-reviewed journal. The eligibility period is the four calendar
years preceding the year of the award. For more information on
the prize, including a list of previous recipients, see www.siam.org/
Eligibility: Candidate papers must be published in English in a peer-
reveiwed journal bearing a publication date in the 2007-2010 cal-
endar years (January I, 2007-December 31, 2010). They must con-
tain significant research contributions to the field of optimization,
as commonly defined in the mathematical literature, with direct or
potential applications.
Description of the award: The award will consist of a plaque and a
certificate containing the citation. At least one of the prize recipi-
ents is expected to attend the award ceremony and to present the
paper at the conference.
Nominations: Nominations, together with a PDF version of the pa-
per, should be submitted via e-mail to one of the members of the
SIAG/OPT Prize Committee before November 15, 2010.
Selection Committee: The members of the selection committee are:
Yinyu Ye (Chair), Stanford University; Shabbir Ahmed, Georgia In-
stitute of Technology; Philip Gill, University of California, San Diego;

Application for Membership
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Etienne de Klerk, Tilburg University, The Netherlands; Jean Philippe
Richard, University of Florida.


The 14th Conference on Integer

Programming and Combinatorial


June 9-11, 2010, Ecole Polytechnique F6d6rale de Lausanne, Switzerland.
The IPCO conference is under the auspices of the Mathematical
Programming Society. It is held every year, except for those years in
which the 'Symposium on Mathematical Programming' takes place.
The conference is meant to be a forum for researchers and prac-
titioners working on various aspects of integer programming and
combinatorial optimization. The aim is to present recent develop-
ments in theory, computation, and applications in these areas.

Mail to:
personal use Mathematical Programming Society
3600 University City Sciences Center
Philadelphia, PA 19104-2688

Cheques or money orders should be made
payable to The Mathematical Programming
Society, Inc. Dues for 2010, including sub-
scription to the journal Mathematical Pro-
gramming, are US $90. Retired are $45.
Student applications: Dues are $22.50.
Have a faculty member verify your student
status and send application with dues to
above address.

Faculty verifying status


Editor: Andrea Lodi, DEIS University of Bologna, Viale Risorgimento 2, 1-40136 Bologna, Italy. andrea.lodi@unibo.it
Co-Editors: Alberto Caprara, DEIS University of Bologna, Viale Risorgimento 2, 1-40136 Bologna, Italy. alberto.caprara@unibo.it
Katya Scheinberg, Department of Industrial Engineering and Operations Research, Columbia University, 500 W 120th Street, New York, NY, 10027.
Founding Editor: Donald W. Hearn

Published by the Mathematical Programming Society.
Design and typesetting by Christoph Eyrich, Mehringdamm 57 / Hof 3, 10961 Berlin, Germany. optima@0x45.de
Printed by Oktoberdruck AG, Berlin.


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