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Population Monotonic Allocation Schemes for Vertex Cover Games

by   Han Xiao, et al.
Ocean University of China

For the class of vertex cover games (introduced by Deng et al., Math. Oper. Res., 24:751-766, 1999), we investigate the population monotonic allocation schemes (introduced by Sprumont, Games Econ. Behav., 2: 378-394, 1990). We present a complete characterization for the class of vertex cover games admitting a population monotonic allocation scheme (PMAS for short), i.e., a vertex cover game has a PMAS if and only if the underlying graph is (K_3,C_4,P_5)-free. Our characterization implies that the existence of a PMAS can be determined efficiently for vertex cover games. We also propose an alternative description for PMAS-es in vertex cover games based on the dual linear program of the vertex cover problem, which reveals the dual-based allocation scheme nature of PMAS-es. Moreover, we give a complete characterization for integral PMAS-es in vertex cover games via stable matchings and show that the celebrated Gale-Shapley algorithm (introduced by Gale and Shapley, Amer. Math. Monthly, 69:9-15, 1962) can be used to produce all integral PMAS-es in vertex cover games.


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