Abstract
We develop for set cover games several general cost-sharing methods that are approximately budget-balanced, in the core, and/or group-strategyproof. We first study the cost sharing for a single set cover game, which does not have a budget-balanced mechanism in the core. We show that there is no cost allocation method that can always recover more than \(\frac{1}{\ln n}\) of the total cost and in the core. Here n is the number of all players to be served. We give a cost allocation method that always recovers \(\frac{1}{\ln d_{\mathit{max}}}\) of the total cost, where d max is the maximum size of all sets. We then study the cost allocation scheme for all induced subgames. It is known that no cost sharing scheme can always recover more than \(\frac{1}{n}\) of the total cost for every subset of players. We give an efficient cost sharing scheme that always recovers at least \(\frac{1}{2n}\) of the total cost for every subset of players and furthermore, our scheme is cross-monotone. When the elements to be covered are selfish agents with privately known valuations, we present a strategyproof charging mechanism, under the assumption that all sets are simple sets; further, the total cost of the set cover is no more than ln d max times that of an optimal solution. When the sets are selfish agents with privately known costs, we present a strategyproof payment mechanism to them. We also show how to fairly share the payments to all sets among the elements.
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The preliminary version of the paper was published at STACS 2005 [19].
The research of Xiang-Yang Li is partially supported by NSF CNS-0832120, NSF CCF-0515088, National Natural Science Foundation of China under Grant No. 60828003, National Basic Research Program of China (973 Program) under grant No. 2006CB30300, the National High Technology Research and Development Program of China (863 Program) under grant No. 2007AA01Z180, Hong Kong RGC HKUST 6169/07, the RGC under Grant HKBU 2104/06E. Li and Lou is also partially supported by CERG under Grant PolyU-5232/07E.
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Li, XY., Sun, Z., Wang, W. et al. Cost sharing and strategyproof mechanisms for set cover games. J Comb Optim 20, 259–284 (2010). https://doi.org/10.1007/s10878-009-9209-x
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DOI: https://doi.org/10.1007/s10878-009-9209-x