Strategyproof cost-sharing Mechanisms for Set Cover and Facility Location Problems

Achieving truth-revealing, also called strategyproofness or incentive compatibility, is fundamental to game theory. The seminal work of Vickery showed a way of achieving this for single item auctions. An extension of Vickery’s idea generales the framework of auctions to the design of strategyproof mechanisms. This goes under the name of Vickery-Clarke-Groves (VCG) mechanism, and is perhaps the single most influential idea in mechanism design. VCG mechanisms inherently fit into the realm of optimization problems and hence give rise to inherent algorithmic issues – this insight was provided by the recent important paper of Nisan and Ronen. Indeed, they even dealt with situations in which the underlying optimization problems were NP-hard, by resorting to methods from the field of approximation algorithms. Strategyproofness also plays an important role in cooperative game theory – when the cost of a shared resource is to be distributed among its users in such a way that revealing true utility is a dominant strategy of users. Much work has been done on obtaining strategyproof cost sharing mechanisms – for instance for the spanning tree game [Bir76, GH81, GH84, KSK96, KSK97]. Once again, the underlying optimization problems of some of the interesting games are NP-hard, and strategyproof cost allocation for several such games have been studied in [FPS00, GS00, JV01, RT00], again using methods from approximation algorithms. In this paper, we obtain strategyproof cost allocations for two fundamental games whose underlying optimization problems are NP-hard, the set cover game and the facility location game. For the latter game, this is made possible by new approximation algorithms for the underlying optimization problem using the technique of dual-fitting [JMM02]. In retrospect, the natural greedy algorithm for the set cover problem (see [Vaz01]) can also analyzed using this technique – we utilize this viewpoint for handling the set cover game. The facility location game was studied in [JV01, GS00], who left the open problem of obtaining a strategyproof mechanism based on a constant factor approximation algorithm. Our paper answers this question