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Complexity in Cooperative Game Theory
"... We introduce cooperative games (N, v) with a description polynomial in n, where n is the number of players. In order to study the complexity of cooperative game problems, we assume that a cooperative game v :2 N ! Q is given by an oracle returning v (S) for each query S N: Finally, we consider sever ..."
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Cited by 11 (0 self)
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We introduce cooperative games (N, v) with a description polynomial in n, where n is the number of players. In order to study the complexity of cooperative game problems, we assume that a cooperative game v :2 N ! Q is given by an oracle returning v (S) for each query S N: Finally, we consider several cooperative game problems and we give a list of complexity results.
On The Computation Of The Nucleolus Of A Cooperative Game
, 1999
"... We consider classes of cooperative games. We show that we can efficiently compute an allocation in the intersection of the prekernel and the least core of the game if we can efficiently compute the minimum excess for any given allocation. In the case where the prekernel of the game contains exactly ..."
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Cited by 11 (1 self)
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We consider classes of cooperative games. We show that we can efficiently compute an allocation in the intersection of the prekernel and the least core of the game if we can efficiently compute the minimum excess for any given allocation. In the case where the prekernel of the game contains exactly one core vector, our algorithm computes the nucleolus of the game. This generalizes both a recent result by Kuipers on the computation of the nucleolus for convex games and a classical result by Megiddo on the nucleolus of standard tree games to classes of more general minimum cost spanning tree games. Our algorithm is based on the ellipsoid method and Maschler's scheme for approximating the prekernel.
An efficient algorithm for nucleolus and prekernel computation in some classes of TUgames
, 1998
"... We consider classes of TUgames. We show that we can efficiently compute an allocation in the intersection of the prekernel and the least core of the game if we can efficiently compute the minimum excess for any given allocation. In the case where the prekernel of the game contains exactly one core ..."
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Cited by 7 (1 self)
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We consider classes of TUgames. We show that we can efficiently compute an allocation in the intersection of the prekernel and the least core of the game if we can efficiently compute the minimum excess for any given allocation. In the case where the prekernel of the game contains exactly one core vector, our algorithm computes the nucleolus of the game. This generalizes both a recent result by Kuipers on the computation of the nucleolus for convex games and a classical result by Megiddo on the nucleolus of standard tree games to classes of more general minimum cost spanning tree games. Our algorithm is based on the ellipsoid method and Maschler's scheme for approximating the prekernel.
Submodularity of Some Classes of Combinatorial Optimization Games
, 2002
"... Some situations concerning cost allocation are formulated as combinatorial optimization games. We consider a minimum coloring game and a minimum vertex cover game. For a minimum coloring game, DengIbarakiNagamochi [1] showed that deciding the core nonemptiness of a given minimum coloring game is N ..."
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Cited by 4 (4 self)
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Some situations concerning cost allocation are formulated as combinatorial optimization games. We consider a minimum coloring game and a minimum vertex cover game. For a minimum coloring game, DengIbarakiNagamochi [1] showed that deciding the core nonemptiness of a given minimum coloring game is NPcomplete, which implies that a good characterization of balanced minimum coloring games is unlikely to be obtained and DengIbarakiNagamochiZang [2] showed that a minimum coloring game is totally balanced if and only if the underlying graph is perfect. For a minimum vertex cover game, DengIbarakiNagamochi [1] showed that a minimum vertex cover game has the nonempty core if and only if the size of a minimum vertex cover of the underlying graph is equal to the size of a maximum matching of the graph, and DengIbarakiNagamochiZang [2] showed that a minimum vertex cover game is totally balanced if and only if the underlying graph is bipartite. In this note, we characterize submodular minimum coloring games and submodular vertex cover games in terms of forbidden subgraphs. That is, a minimum coloring game is submodular if and only if the underlying graph contains no induced subgraph isomorphic to K and a minimum vertex cover game is submodular if and only if the underlying graph contains no subgraph isomorphic to P³ or K³. A relationship with matroids is also stated.
Traveling salesman games with the Monge property
, 2001
"... Several works indicate the relationship between wellsolved combinatorial optimization problems and the core nonemptiness of cooperative games associated with them. In this note, we consider the core of traveling salesman games. We show that the core of traveling salesman games in which the dist ..."
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Cited by 2 (0 self)
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Several works indicate the relationship between wellsolved combinatorial optimization problems and the core nonemptiness of cooperative games associated with them. In this note, we consider the core of traveling salesman games. We show that the core of traveling salesman games in which the distance matrix is a Monge matrix is nonempty. This is the first result for traveling salesman games related with a wellsolved structure. Moreover, we show that the problem of testing the core nonemptiness of a given traveling salesman game is NPhard.
Fair Cost Allocations under Conflicts  A Gametheoretic Point of View
, 2003
"... We study the cost allocation problem when the players are involved in a conflict situation. More formally, we consider a minimum coloring game, introduced by Deng, Ibaraki & Nagamochi, and provide algorithms for the core, the #value, the nucleolus and the Shapley value on some classes of graphs ..."
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We study the cost allocation problem when the players are involved in a conflict situation. More formally, we consider a minimum coloring game, introduced by Deng, Ibaraki & Nagamochi, and provide algorithms for the core, the #value, the nucleolus and the Shapley value on some classes of graphs. The investigation gives several insights to the relationship of algorithm theory with cooperative games.
Computing Shapley Value in Supermodular
"... Abstract. Coalitional games allow subsets (coalitions) of players to cooperate to receive a collective payoff. This payoff is then distributed “fairly ” among the members of that coalition according to some division scheme. Various solution concepts have been proposed as reasonable schemes for gener ..."
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Abstract. Coalitional games allow subsets (coalitions) of players to cooperate to receive a collective payoff. This payoff is then distributed “fairly ” among the members of that coalition according to some division scheme. Various solution concepts have been proposed as reasonable schemes for generating fair allocations. The Shapley value is one classic solution concept: player i’s share is precisely equal to i’s expected marginal contribution if the players join the coalition one at a time, in a uniformly random order. In this paper, we consider the class of supermodular games (sometimes called convex games), define and survey computational results on other standard solution concepts, and contrast these results with new results regarding the Shapley value. In particular, we give a fully polynomialtime randomized approximation scheme (FPRAS) to compute the Shapley value to within a (1 ± ε) factor in monotone supermodular games. We show that this result is tight in several senses: no deterministic algorithm can approximate Shapley value as well, no randomized algorithm can do better, and both monotonicity and supermodularity are required for the existence of an efficient (1 ± ε)approximation algorithm. We also argue that, relative to supermodularity, monotonicity is a mild assumption, and we discuss how to transform supermodular games to be monotonic. Topic classification: algorithmic game theory, algorithms, computational complexity