Optimal Efficiency Guarantees for Network Design Mechanisms

A cost-sharing problem is defined by a set of players vying to receive some good or service, and a cost function describing the cost incurred by the auctioneer as a function of the set of winners. A cost-sharing mechanism is a protocol that decides which players win the auction and at what prices. Three desirable but provably 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 good; budget-balance, meaning that the mechanism recovers its incurred cost with the prices charged; and efficiency, meaning that the cost incurred and the value to the players served are traded off in an optimal way. Our work is motivated by the following fundamental question: for which cost-sharing problems are incentive-compatible mechanisms with good approximate budget-balance and efficiency possible? We focus on cost functions defined implicitly by NP-hard combinatorial optimization problems, including the metric uncapacitated facility location problem, the Steiner tree problem, and rent-or-buy network design problems. For facility location and rent-or-buy network design, we establish for the first time that approximate budget-balance and efficiency are simultaneously possible. For the Steiner tree problem, where such a guarantee was previously known, we prove a new, optimal lower bound on the approximate efficiency achievable by the wide and natural class of "Moulin mechanisms". This lower bound exposes a latent approximation hierarchy among different cost-sharing problems.

[1]  Anna R. Karlin,et al.  Optimization in the private value model: competitive analysis applied to auction design , 2003 .

[2]  Vijay V. Vazirani,et al.  Approximation Algorithms , 2001, Springer Berlin Heidelberg.

[3]  William Vickrey,et al.  Counterspeculation, Auctions, And Competitive Sealed Tenders , 1961 .

[4]  Vijay V. Vazirani,et al.  Applications of approximation algorithms to cooperative games , 2001, STOC '01.

[5]  Tim Roughgarden,et al.  Approximation via cost-sharing: a simple approximation algorithm for the multicommodity rent-or-buy problem , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[6]  Ilya Segal,et al.  Solutions manual for Microeconomic theory : Mas-Colell, Whinston and Green , 1997 .

[7]  Éva Tardos,et al.  Cost Sharing and Approximation , 2005 .

[8]  Nikhil R. Devanur,et al.  Strategyproof cost-sharing mechanisms for set cover and facility location games , 2003, EC '03.

[9]  Ariel Rubinstein,et al.  A Course in Game Theory , 1995 .

[10]  H. Moulin Incremental cost sharing: Characterization by coalition strategy-proofness , 1999 .

[11]  Tim Roughgarden,et al.  Approximately Efficient Cost-Sharing Mechanisms , 2006, ArXiv.

[12]  Jochen Könemann,et al.  Simple cost sharing schemes for multicommodity rent-or-buy and stochastic Steiner tree , 2006, STOC '06.

[13]  Stefano Leonardi,et al.  Cross-monotonic cost-sharing methods for connected facility location games , 2004, EC '04.

[14]  R. Ravi,et al.  An efficient cost-sharing mechanism for the prize-collecting Steiner forest problem , 2007, SODA '07.

[15]  Joan Feigenbaum,et al.  Hardness Results for Multicast Cost Sharing , 2002, FSTTCS.

[16]  Joan Feigenbaum,et al.  Distributed algorithmic mechanism design: recent results and future directions , 2002, DIALM '02.

[17]  David P. Williamson,et al.  A note on the prize collecting traveling salesman problem , 1993, Math. Program..

[18]  Joan Feigenbaum,et al.  Sharing the Cost of Multicast Transmissions , 2001, J. Comput. Syst. Sci..

[19]  Aravind Srinivasan,et al.  Cost-Sharing Mechanisms for Network Design , 2007, Algorithmica.

[20]  David R. Karger,et al.  Building Steiner trees with incomplete global knowledge , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[21]  Tim Roughgarden,et al.  Simpler and better approximation algorithms for network design , 2003, STOC '03.

[22]  Tim Roughgarden,et al.  New trade-offs in cost-sharing mechanisms , 2006, STOC '06.

[23]  Luca Becchetti,et al.  Sharing the cost more efficiently: improved approximation for multicommodity rent-or-buy , 2005, SODA '05.

[24]  Jochen Könemann,et al.  A group-strategyproof mechanism for Steiner forests , 2005, SODA '05.

[25]  J. Laffont Aggregation and revelation of preferences , 1979 .

[26]  Éva Tardos,et al.  Group strategy proof mechanisms via primal-dual algorithms , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[27]  Jochen Könemann,et al.  From Primal-Dual to Cost Shares and Back: A Stronger LP Relaxation for the Steiner Forest Problem , 2005, ICALP.

[28]  Aravind Srinivasan,et al.  Cost-Sharing Mechanisms for Network Design , 2004, APPROX-RANDOM.

[29]  Robin Milner,et al.  On Observing Nondeterminism and Concurrency , 1980, ICALP.

[30]  Jerry R. Green,et al.  Partial Equilibrium Approach to the Free-Rider Problem , 1976 .

[31]  Nicole Immorlica,et al.  Limitations of cross-monotonic cost sharing schemes , 2005, SODA '05.

[32]  Tim Roughgarden,et al.  Optimal Cost-Sharing Mechanisms for Steiner Forest Problems , 2006, WINE.

[33]  A. Mas-Colell,et al.  Microeconomic Theory , 1995 .

[34]  H. Moulin,et al.  Strategyproof sharing of submodular costs:budget balance versus efficiency , 2001 .

[35]  Vijay V. Vazirani,et al.  Equitable cost allocations via primal-dual-type algorithms , 2002, STOC '02.

[36]  Joan Feigenbaum,et al.  Approximation and collusion in multicast cost sharing , 2003, EC '03.