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On the impact of combinatorial structure on congestion games
 FOCS
"... We study the impact of combinatorial structure in congestion games on the complexity of computing pure Nash equilibria and the convergence time of best response sequences. In particular, we investigate which properties of the strategy spaces of individual players ensure a polynomial convergence time ..."
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Cited by 51 (14 self)
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We study the impact of combinatorial structure in congestion games on the complexity of computing pure Nash equilibria and the convergence time of best response sequences. In particular, we investigate which properties of the strategy spaces of individual players ensure a polynomial convergence time. We show that if the strategy space of each player consists of the bases of a matroid over the set of resources, then the lengths of all best response sequences are polynomially bounded in the number of players and resources. We also prove that this result is tight, that is, the matroid property is a necessary and sufficient condition on the players ’ strategy spaces for guaranteeing polynomial time convergence to a Nash equilibrium. In addition, we present an approach that enables us to devise hardness proofs for various kinds of combinatorial games, including first results about the hardness of market sharing games and congestion games for overlay network design. Our approach also yields a short proof for the PLScompleteness of network congestion games. In particular, we show that network congestion games are PLScomplete for directed and undirected networks even in case of linear latency functions.
Tight approximation algorithms for maximum general assignment problems
 Proc. of ACMSIAM SODA
, 2006
"... A separable assignment problem (SAP) is defined by a set of bins and a set of items to pack in each bin; a value, fij, for assigning item j to bin i; and a separate packing constraint for each bin – i.e. for bin i, a family Ii of subsets of items that fit in bin i. The goal is to pack items into bin ..."
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Cited by 43 (8 self)
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A separable assignment problem (SAP) is defined by a set of bins and a set of items to pack in each bin; a value, fij, for assigning item j to bin i; and a separate packing constraint for each bin – i.e. for bin i, a family Ii of subsets of items that fit in bin i. The goal is to pack items into bins to maximize the aggregate value. This class of problems includes the maximum generalized assignment problem (GAP) 1) and a distributed caching problem (DCP) described in this paper. Given a βapproximation algorithm for finding the highest value packing of a single bin, we give 1. A polynomialtime LProunding based ((1 − 1 e)β)approximation algorithm. 2. A simple polynomialtime local search ( β approximation algorithm, for any ɛ> 0. β+1 − ɛ)Therefore, for all examples of SAP that admit an approximation scheme for the singlebin problem, we obtain an LPbased algorithm with (1 − 1 e − ɛ)approximation and a local search algorithm with ( 1 2 −ɛ)approximation guarantee. Furthermore, for cases in which the subproblem admits a fully polynomial approximation scheme (such as for GAP), the LPbased algorithm analysis can be strengthened to give a guarantee of 1 − 1 e. The best previously known approximation algorithm for GAP is a 1 2approximation by Shmoys and Tardos; and Chekuri and Khanna. Our LP algorithm is based on rounding a new linear programming relaxation, with a provably better integrality gap. To complement these results, we show that SAP and DCP cannot be approximated within a factor better than 1 − 1 e unless NP ⊆ DTIME(n O(log log n)), even if there exists a polynomialtime exact algorithm for the singlebin problem.
A Unified Approach to Congestion Games and TwoSided Markets
 In Proceedings of the 3nd International Workshop on Internet and Network Economics (WINE 2007
, 2007
"... Congestion games are a wellstudied model for resource sharing among uncoordinated selfish agents. Usually, one assumes that the resources in a congestion game do not have any preferences over the players that can allocate them. In typical load balancing applications, however, different jobs can hav ..."
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Cited by 6 (5 self)
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Congestion games are a wellstudied model for resource sharing among uncoordinated selfish agents. Usually, one assumes that the resources in a congestion game do not have any preferences over the players that can allocate them. In typical load balancing applications, however, different jobs can have different priorities, and jobs with higher priorities get, for example, larger shares of the processor time. We introduce a model in which each resource can assign priorities to the players and players with higher priorities can displace players with lower priorities. Our model does not only extend standard congestion games, but it can also be seen as a model of twosided markets with ties. We prove that singleton congestion games with priorities are potential games, and we show that every playerspecific singleton congestion game with priorities possesses a pure Nash equilibrium that can be found in polynomial time. Finally, we extend our results to matroid congestion games, in which the strategy space of each player consists of the bases of a matroid over the resources.
Anarchy, stability, and utopia: Creating better matchings. Autonomous Agents and MultiAgent Systems
, 2011
"... We consider the loss in social welfare caused by individual rationality in matching scenarios. We give both theoretical and experimental results comparing stable matchings with socially optimal ones, as well as studying the convergence of various natural algorithms to stable matchings. Our main goal ..."
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Cited by 5 (3 self)
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We consider the loss in social welfare caused by individual rationality in matching scenarios. We give both theoretical and experimental results comparing stable matchings with socially optimal ones, as well as studying the convergence of various natural algorithms to stable matchings. Our main goal is to design mechanisms in order to incentivize agents to participate in matchings that are socially desirable. We show that theoretically, the loss in social welfare caused by strategic behavior can be substantial. We analyze some natural distributions of utilities that agents receive from matchings, and find that in most cases the stable matching attains close to the optimal social welfare. Furthermore, for certain graph structures, simple greedy algorithms for partnerswitching (some without convergence guarantees) converge to stability remarkably quickly in expectation. Even when stable matchings are significantly socially suboptimal, slight changes in incentives can provide good solutions. We derive conditions for the existence of approximately stable matchings that are also close to socially optimal, which demonstrates that adding small switching costs can make socially optimal matchings stable. We also show that introducing heterogeneity in tastes can greatly improve social outcomes. 1.
Uncoordinated TwoSided Markets
, 2007
"... Various economic interactions can be modeled as twosided markets. A central solution concept to these markets are stable matchings, introduced by Gale and Shapley. It is well known that stable matchings can be computed in polynomial time, but many reallife markets lack a central authority to match ..."
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Cited by 4 (1 self)
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Various economic interactions can be modeled as twosided markets. A central solution concept to these markets are stable matchings, introduced by Gale and Shapley. It is well known that stable matchings can be computed in polynomial time, but many reallife markets lack a central authority to match agents. In those markets, matchings are formed by actions of selfinterested agents. Knuth introduced uncoordinated twosided markets and showed that the uncoordinated better response dynamics may cycle. However, Roth and Vande Vate showed that the random better response dynamics converges to a stable matching with probability one, but did not address the question of convergence time. In this paper, we give an exponential lower bound for the convergence time of the random better response dynamics in twosided markets. We also extend these results to the best response dynamics, i. e., we present a cycle of best responses, and prove that the random best response dynamics converges to a stable matching with probability one, but its convergence time is exponential. Additionally, we identify the special class of correlated twosided markets with reallife applications for which we prove that the random best response dynamics converges in expected polynomial time.
Uncoordinated TwoSided Matching Markets
 EC'08
, 2008
"... Various economic interactions can be modeled as twosided markets. A central solution concept to these markets are stable matchings, introduced by Gale and Shapley. It is well known that stable matchings can be computed in polynomial time, but many reallife markets lack a central authority to match ..."
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Cited by 3 (0 self)
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Various economic interactions can be modeled as twosided markets. A central solution concept to these markets are stable matchings, introduced by Gale and Shapley. It is well known that stable matchings can be computed in polynomial time, but many reallife markets lack a central authority to match agents. In those markets, matchings are formed by actions of selfinterested agents. Knuth introduced uncoordinated twosided markets and showed that the uncoordinated better response dynamics may cycle. However, Roth and Vande Vate showed that the random better response dynamics converges to a stable matching with probability one, but did not address the question of convergence time. In this paper, we give an exponential lower bound for the convergence time of the random better response dynamics in twosided markets. We also extend the results for the better response dynamics to the best response dynamics, i.e., we present a cycle of best responses, and prove that the random best response dynamics converges to a stable matching with probability one, but its convergence time is exponential. Additionally, we identify the special class of correlated matroid twosided markets with reallife applications for which we prove that the random best response dynamics converges in expected polynomial time.
Tight Approximation Algorithms for Maximum Separable Assignment Problems
"... A separable assignment problem (SAP) is defined by a set of bins and a set of items to pack in each bin; a value, fij, for assigning item j to bin i; and a separate packing constraint for each bin—i.e., for each bin, a family of subsets of items that fit in to that bin. The goal is to pack items int ..."
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Cited by 1 (0 self)
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A separable assignment problem (SAP) is defined by a set of bins and a set of items to pack in each bin; a value, fij, for assigning item j to bin i; and a separate packing constraint for each bin—i.e., for each bin, a family of subsets of items that fit in to that bin. The goal is to pack items into bins to maximize the aggregate value. This class of problems includes the maximum generalized assignment problem (GAP) 1 and a distributed caching problem (DCP) described in this paper. Given a �approximation algorithm for finding the highest value packing of a single bin, we give (i) A polynomialtime LProunding based ��1 − 1/e���approximation algorithm. (ii) A simple polynomialtime local search ��/� � + 1 � − ��approximation algorithm, for any �> 0. Therefore, for all examples of SAP that admit an approximation scheme for the singlebin problem, we obtain an LPbased algorithm with �1 − 1/e − ��approximation and a local search algorithm with � 1 − ��approximation guarantee. Furthermore, 2 for cases in which the subproblem admits a fully polynomial approximation scheme (such as for GAP), the LPbased algorithm analysis can be strengthened to give a guarantee of 1 − 1/e. The best previously known approximation algorithm for GAP is a 1approximation by Shmoys and Tardos and Chekuri and Khanna. Our LP algorithm is based on rounding a 2
c○20xx INFORMS Tight Approximation Algorithms for Maximum Separable Assignment Problems
"... A separable assignment problem (SAP) is defined by a set of bins and a set of items to pack in each bin; a value, fij, for assigning item j to bin i; and a separate packing constraint for each bin – i.e. for bin i, a family Ii of subsets of items that fit in bin i. The goal is to pack items into bin ..."
Abstract
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A separable assignment problem (SAP) is defined by a set of bins and a set of items to pack in each bin; a value, fij, for assigning item j to bin i; and a separate packing constraint for each bin – i.e. for bin i, a family Ii of subsets of items that fit in bin i. The goal is to pack items into bins to maximize the aggregate value. This class of problems includes the maximum generalized assignment problem (GAP) 1) and a distributed caching problem (DCP) described in this paper. Given a βapproximation algorithm for finding the highest value packing of a single bin, we give (i) A polynomialtime LProunding based ((1 − 1)β)approximation algorithm. e (ii) A simple polynomialtime local search ( β − ɛ)approximation algorithm, for any ɛ> 0. β+1 Therefore, for all examples of SAP that admit an approximation scheme for the singlebin problem, we obtain an LPbased algorithm with (1 − 1 1 − ɛ)approximation and a local search algorithm with ( − ɛ)approximation guarantee. Furthermore, e 2
Noname manuscript No. (will be inserted by the editor) Anarchy, Stability, and Utopia: Creating Better Matchings
, 2011
"... Abstract Historically, the analysis of matching has centered on designing algorithms to produce stable matchings as well as on analyzing the incentive compatibility of matching mechanisms. Less attention has been paid to questions related to the social welfare of stable matchings in cardinal utility ..."
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Abstract Historically, the analysis of matching has centered on designing algorithms to produce stable matchings as well as on analyzing the incentive compatibility of matching mechanisms. Less attention has been paid to questions related to the social welfare of stable matchings in cardinal utility models. We examine the loss in social welfare that arises from requiring matchings to be stable, the natural equilibrium concept under individual rationality. While this loss can be arbitrarily bad under general preferences, when there is some structure to the underlying graph corresponding to natural conditions on preferences, we prove worst case bounds on the price of anarchy. Surprisingly, under simple distributions of utilities, the average case loss turns out to be significantly smaller than the worstcase analysis would suggest. Furthermore, we derive conditions for the existence of approximately stable matchings that are also close to socially optimal, demonstrating that adding small switching costs can make socially (near)optimal matchings stable. Our analysis leads to several concomitant results of interest on the convergence of decentralized partnerswitching algorithms, and on the impact of heterogeneity of tastes on social welfare.