Approximately Efficient Cost-Sharing Mechanisms

We study cost-sharing mechanisms for several fundamental NP-hard combinatorial optimization problems. A cost-sharing mechanism is a protocol that, given bids for a service, determines which bidders to serve and what prices to charge. The mechanism incurs a subset-dependent cost that is implicitly defined by an instance of a combinatorial optimization problem. Three desirable but mutually incompatible properties of a cost-sharing mechanism are: incentive-compatibility, meaning that players are motivated to bid their true private value for receiving the service; budget-balance, meaning that the mechanism recovers its incurred cost with the prices charged; and efficiency, meaning that the cost incurred and the valuations of the players served are traded off in an optimal way. Our work is motivated by the following fundamental question: for which cost functions, and in what senses, are incentive-compatible mechanisms with good approximate budget-balance and efficiency possible? We make three different types of contributions to this question. • We identify several new classes of combinatorial cost functions that admit incentive-compatible mechanisms achieving both a constant-factor approximation of budget-balance and a polylogarithmic approximation of the social cost formulation of efficiency. In particular, we exhibit such mechanisms for the classes of facility location and single-sink rent-or-buy cost functions, with approximate efficiency O(log k) and O(log k), respectively. (Here k denotes the number of players served in an efficient solution.) The mechanisms belong to the class of Moulin mechanisms, and therefore satisfy a strong version of incentive-compatibility known as groupstrategyproofness. • We prove a new, optimal lower bound of Ω(log k) on the approximate efficiency of every O(1)budget-balanced Moulin mechanism for Steiner tree or SSRoB cost functions. This lower bound exposes a latent approximation hierarchy among different cost-sharing problems. • We show that weakening the definition of incentive-compatibility to strategyproofness can permit exponentially more efficient approximately budget-balanced mechanisms, in particular for set cover cost-sharing problems. Department of Computer Science, Stanford University, 462 Gates Building, 353 Serra Mall, Stanford, CA 94305. Supported in part by ONR grant N00014-04-1-0725, DARPA grant W911NF-05-1-0224, and an NSF CAREER Award. Email: [email protected]. Department of Computer Science, Stanford University, 470 Gates Building, 353 Serra Mall, Stanford, CA 94305. Supported in part by OSD/ONR CIP/SW URI ”Software Quality and Infrastructure Protection for Diffuse Computing” through ONR Grant N00014-01-1-0795 and by OSD/ONR CIP/SW URI ”Trustworthy Infrastructure, Mechanisms, and Experimentation for Diffuse Computing” through ONR Grant N00014-04-1-0725. Email: [email protected].