Chapter

Computing and Combinatorics

Volume 2108 of the series Lecture Notes in Computer Science pp 247-256

Date:

Membership for Core of LP Games and Other Games

  • Qizhi FangAffiliated withDepartment of Applied Mathematics, Qingdao Ocean University
  • , Shanfeng ZhuAffiliated withDepartment of Computer Science, City University of Hong Kong
  • , Maocheng CaiAffiliated withInstitute of Systems Science, Chinese Academy of Sciences
  • , Xiaotie DengAffiliated withDepartment of Computer Science, City University of Hong Kong

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Abstract

Let Γ = (N, v) be a cooperative game with the player set N and characteristic function v: 2NR. An imputation of the game is in the core if no subset of players could gain advantage by splitting from the grand coalition of all players. It is well known that, for the linear production game, and the flow game, the core is always non-empty (and a solution in the core can be found in polynomial time). In this paper, we show that, given an imputation x, it is NP-complete to decide it is not a member of the core, in both games. The same also holds for Steiner tree game. In addition, for Steiner tree games, we prove that testing the total balacedness is NP-hard.

Key words

cooperative game core network flow linear programming Steiner tree NP-completeness