Fang Q., Zhu S., Cai M., Deng X. (2001) Membership for Core of LP Games and Other Games. In: Wang J. (eds) Computing and Combinatorics. COCOON 2001. Lecture Notes in Computer Science, vol 2108. Springer, Berlin, Heidelberg
Let Γ = (N, v) be a cooperative game with the player set N and characteristic function v: 2N → R. 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.
cooperative game core network flow linear programming Steiner tree NP-completeness