Abstract
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.
Research is partially supported by a grant from the Research Grants Council of Hong Kong SAR(Cit yU 1116/99E) and a grant from CityU of Hong Kong (Project No. 7001215).
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
I.J. Curiel (1988), Cooperative Game Theory and Applications. Ph.D. dissertation, University of Nijmegen, the Netherlands.
X. Deng, T. Ibaraki and H. Nagamochi (1999), Algorithmic aspects of the core of combinatorial optimization games. Mathematics of Operations Research 24, pp. 751–766.
X. Deng, T. Ibaraki, H. Nagamochi and W. Zang (2000),Totally balanced combinatorial optimization games, Mathematical Programming 87, pp.441–452.
X. Deng and C.H. Papadimitriou (1994), On the complexity of cooperative solution concepts. Mathematics of Operations Research 19, pp. 257–266.
U. Faigle, W. Kern, S.P. Fekete and W. Hochstättler (1997), On the complexity of testing membership in the core of min-cost spanning tree games. International Journal of Game Theory 26, pp. 361–366.
M.R. Garey and D.S. Johnson (1979), Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, San Francisco.
M.X. Goemans and M. Skutella (2000), Cooperative facility location games. SODA.
D. Granot and G. Huberman (1981), Minimum cost spanning tree games. Mathematical Programming 21, pp. 1–18.
E. Kalai and E. Zemel (1982), Totally balanced games and games of flow. Mathematics of Operations Research 7, pp. 476–478.
E. Kalai and E. Zemel (1982), Generalized network problems yielding totally balanced games. Operations Research 30, pp. 498–1008.
N. Megiddo (1978), Computational complexity and the game theory approach to cost allocation for a tree. Mathematics of Operations Research 3, pp. 189–196.
G. Owen (1975), On the core of linear production games. Mathematical Programming 9, pp. 358–370.
L.S. Shapley (1967), On balanced sets and cores. Naval Research Logistics Quarterly 14, pp. 453–460.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
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. https://doi.org/10.1007/3-540-44679-6_27
Download citation
DOI: https://doi.org/10.1007/3-540-44679-6_27
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42494-9
Online ISBN: 978-3-540-44679-8
eBook Packages: Springer Book Archive