Clique polynomials and independent set polynomials of graphs
Selected papers of the 13th British Combinatorial Conference on British combinatorial conference
On the complexity of cooperative solution concepts
Mathematics of Operations Research
An algorithm for finding the nucleolus of assignment games
International Journal of Game Theory
Computing the nucleolus of min-cost spanning tree games is NP-hard
International Journal of Game Theory
Algorithmic Aspects of the Core of Combinatorial Optimization Games
Mathematics of Operations Research
On the computation of the nucleolus of a cooperative game
International Journal of Game Theory
The Complexity of Counting in Sparse, Regular, and Planar Graphs
SIAM Journal on Computing
Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57)
Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57)
Combinatorica
Core Stability of Minimum Coloring Games
Mathematics of Operations Research
Counting the number of independent sets in chordal graphs
Journal of Discrete Algorithms
On the cores of games arising from integer edge covering functions of graphs
Journal of Combinatorial Optimization
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Optimization theory resolves problems to minimize total costs when the agents are involved in some conflicts. In this paper, we consider how to allocate the minimized total cost among the agents. To do that, the allocation is required to be fair in a certain sense. We use a game-theoretic point of view, and provide algorithms to compute fair allocations in polynomial time for a certain conflict situation. More specifically, we study a minimum coloring game, introduced by Deng, Ibaraki and Nagamochi [X. Deng, T. Ibaraki, H. Nagamochi, Algorithmic aspects of the core of combinatorial optimization games, Math. Oper. Res. 24 (1999) 751-766], and investigate the core, the nucleolus, the @t-value, and the Shapley value. In particular, we provide the following four results. (1) The characterization of the core for a perfect graph in terms of its extreme points. This leads to polynomial-time algorithms to compute a vector in the core, and to determine whether a given vector belongs to the core. (2) A characterization of the nucleolus for some classes of the graphs, including the complete multipartite graphs and the chordal graphs. This leads to a polynomial-time algorithm to compute the nucleolus for these classes of graphs. (3) A polynomial-time algorithm to compute the @t-value for a perfect graph. (4) A polynomial-time algorithm to compute the Shapley value for a forest. The investigation of this paper gives several insights to the relationship of algorithm theory with cooperative games.