Submodularity of minimum-cost spanning tree games

Kobayashi, Makiko; Okamoto, Yoshio

doi:10.1002/net.21540

Cited by 9 publications

(11 citation statements)

References 17 publications

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“…The next lemma extends the scope of Lemma 2 to include violated cycles. When k = 2, this coincides with the condition given by Kobayashi and Okamoto [9] because every cycle in G 1 is well-covered.…”

Section: Definitionsupporting

confidence: 82%

“…The first one, described in Sect. 3, is a generalization of Kobayashi and Okamoto's result [9]. When the edges can have only two distinct weights, they proved that the only obstruction to submodularity comes from the presence of certain cycles in the graph induced by the cheaper edges.…”

Section: ∀A B ⊆ N ν(A) + ν(B) ≥ ν(A ∪ B) + ν(A ∩ B)mentioning

confidence: 85%

“…We conclude our paper with an additional result. As previously mentioned, the authors of [9] gave a necessary condition for submodularity of the spanning tree game. They also stated that they do not know whether their condition can be verified in polynomial time.…”

Section: ∀A B ⊆ N ν(A) + ν(B) ≥ ν(A ∪ B) + ν(A ∩ B)mentioning

confidence: 99%

“…Trudeau [14] gave a sufficient condition for an instance of the game to be submodular. An important step forward was made by Kobayashi and Okamoto [9], who gave a characterization of submodularity for instances of the spanning tree game where the edge weights are restricted to take only two values. For general weights, they stated some necessary (but not always sufficient) as well as some sufficient (but not always necessary) conditions for an instance to be submodular.…”

Section: ∀A B ⊆ N ν(A) + ν(B) ≥ ν(A ∪ B) + ν(A ∩ B)mentioning

confidence: 99%

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An efficient characterization of submodular spanning tree games

Koh

Sanità

2020

Math. Program.

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Cooperative games form an important class of problems in game theory, where a key goal is to distribute a value among a set of players who are allowed to cooperate by forming coalitions. An outcome of the game is given by an allocation vector that assigns a value share to each player. A crucial aspect of such games is submodularity (or convexity). Indeed, convex instances of cooperative games exhibit several nice properties, e.g. regarding the existence and computation of allocations realizing some of the most important solution concepts proposed in the literature. For this reason, a relevant question is whether one can give a polynomial-time characterization of submodular instances, for prominent cooperative games that are in general non-convex. In this paper, we focus on a fundamental and widely studied cooperative game, namely the spanning tree game. An efficient recognition of submodular instances of this game was not known so far, and explicitly mentioned as an open question in the literature. We here settle this open problem by giving a polynomial-time characterization of submodular spanning tree games.

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Section: Definitionsupporting

confidence: 82%

Section: ∀A B ⊆ N ν(A) + ν(B) ≥ ν(A ∪ B) + ν(A ∩ B)mentioning

confidence: 85%

Section: ∀A B ⊆ N ν(A) + ν(B) ≥ ν(A ∪ B) + ν(A ∩ B)mentioning

confidence: 99%

Section: ∀A B ⊆ N ν(A) + ν(B) ≥ ν(A ∪ B) + ν(A ∩ B)mentioning

confidence: 99%

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An efficient characterization of submodular spanning tree games

Koh

Sanità

2020

Math. Program.

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“…Games can be noncooperative or cooperative, each requiring specific mathematical tools. In network analysis, game‐based approaches have been adopted, for example, in congestion , capacitated network design , cost allocation , etc.…”

Section: Introductionmentioning

confidence: 99%

Some properties of transportation network cooperative games

2019

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A basic question in network analysis concerns the quantification of the importance of each node in terms of network connectivity. To this end, a possible approach consists in using cooperative game theory tools to define a measure of node centrality. In this paper, given a transportation network, a cooperative game model with transferable utility (TU game) is considered. The nodes of the network represent the players in such a game, and the Shapley values of the nodes are used to measure centrality. The model, called Transportation Network cooperative (TNc) game, integrates within the utility function of the TU game the network topology and the demand. Properties of TNc games and their associated utility functions are investigated. Numerical results are reported, to get insights into the obtained properties.

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