Inapproximability Results for Combinatorial Auctions with Submodular Utility Functions

Abstract We consider the following allocation problem arising in the setting of combinatorial auctions: a set of goods is to be allocated to a set of players so as to maximize the sum of the utilities of the players (i.e., the social welfare). In the case when the utility of each player is a monotone submodular function, we prove that there is no polynomial time approximation algorithm which approximates the maximum social welfare by a factor better than 1−1/e≃0.632, unless P=NP. Our result is based on a reduction from a multi-prover proof system for MAX-3-COLORING.

[1]  U. Feige,et al.  The allocation problem with submodular utility functions (preliminary version) , 2006 .

[2]  Daniel Lehmann,et al.  Combinatorial auctions with decreasing marginal utilities , 2001, EC '01.

[3]  Ran Raz,et al.  A parallel repetition theorem , 1995, STOC '95.

[4]  Yoav Shoham,et al.  Combinatorial Auctions , 2005, Encyclopedia of Wireless Networks.

[5]  Erez Petrank.  The hardness of approximation: Gap location , 2005, computational complexity.

[6]  Éva Tardos,et al.  An approximate truthful mechanism for combinatorial auctions with single parameter agents , 2003, SODA '03.

[7]  Yoav Shoham,et al.  Truth revelation in approximately efficient combinatorial auctions , 2002, EC '99.

[8]  Uriel Feige,et al.  On maximizing welfare when utility functions are subadditive , 2006, STOC '06.

[9]  Carsten Lund,et al.  Proof verification and the intractability of approximation problems , 1992, FOCS 1992.

[10]  Carsten Lund,et al.  On the hardness of approximating minimization problems , 1993, STOC.

[11]  Noam Nisan,et al.  On the Computational Power of Demand Queries , 2009, SIAM J. Comput..

[12]  Noam Nisan,et al.  Truthful randomized mechanisms for combinatorial auctions , 2006, STOC '06.

[13]  Noam Nisan,et al.  Approximation algorithms for combinatorial auctions with complement-free bidders , 2005, STOC '05.

[14]  Moshe Tennenholtz,et al.  Bundling equilibrium in combinatorial auctions , 2002, Games Econ. Behav..

[15]  Noam Nisan,et al.  The communication requirements of efficient allocations and supporting prices , 2006, J. Econ. Theory.

[16]  Faruk Gul,et al.  WALRASIAN EQUILIBRIUM WITH GROSS SUBSTITUTES , 1999 .

[17]  Tuomas Sandholm,et al.  An algorithm for optimal winner determination in combinatorial auctions , 1999, IJCAI 1999.

[18]  Uriel Feige,et al.  Approximating the domatic number , 2000, STOC '00.

[19]  Yishay Mansour,et al.  Auctions with Budget Constraints , 2004, SWAT.

[20]  Uriel Feige,et al.  Zero knowledge and the chromatic number , 1996, Proceedings of Computational Complexity (Formerly Structure in Complexity Theory).

[21]  Noam Nisan,et al.  Incentive compatible multi unit combinatorial auctions , 2003, TARK '03.

[22]  Aranyak Mehta,et al.  Inapproximability Results for Combinatorial Auctions with Submodular Utility Functions , 2005, Algorithmica.

[23]  Shahar Dobzinski,et al.  An improved approximation algorithm for combinatorial auctions with submodular bidders , 2006, SODA '06.