Core Stability of Vertex Cover Games

In this paper, we focus on the core stability of vertex cover games, which arise from vertex cover problems on graphs. Based on duality theory of linear programming, we first prove that a balanced vertex cover game has the stable core if and only if every edge belongs to a maximum matching in the corresponding graph. We also show that for a totally balanced vertex cover game, the core largeness, extendability and exactness are all equivalent, which imply the core stability.

[1]  Toshihide Ibaraki,et al.  Totally balanced combinatorial optimization games , 2000, Math. Program..

[2]  L. Shapley.  Cores of convex games , 1971 .

[3]  Xiaotie Deng,et al.  On the Complexity of Cooperative Solution Concepts , 1994, Math. Oper. Res..

[4]  T. E. S. Raghavan,et al.  Assignment games with stable core , 2001, Int. J. Game Theory.

[5]  W. Sharkey,et al.  Cooperative games with large cores , 1982 .

[6]  D. R. Fulkerson,et al.  On balanced matrices , 1974 .

[7]  Qizhi Fang,et al.  Core Stability of Flow Games , 2005, CJCDGCGT.

[8]  Bas van Velzen,et al.  Dominating set games , 2004, Oper. Res. Lett..

[9]  J. Neumann,et al.  Theory of games and economic behavior , 1945, 100 Years of Math Milestones.

[10]  Toshihide Ibaraki,et al.  Algorithmic Aspects of the Core of Combinatorial Optimization Games , 1999, Math. Oper. Res..

[11]  Hans Reijnierse,et al.  Prosperity properties of TU-games , 1999, Int. J. Game Theory.

[12]  Yoshio Okamoto,et al.  Core Stability of Minimum Coloring Games , 2004, WG.