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Inapproximability Results for Combinatorial Auctions with Submodular Utility Functions

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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.

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References

  1. Andelman, N., Mansour, Y.: Auctions with budget constraints. In: Scandinavian Workshop on Algorithm Theory, pp. 26–38 (2004)

  2. Archer, A., Papadimitriou, C., Talwar, K., Tardos, E.: An approximate truthful mechanism for combinatorial auctions with single parameter agents. In: ACM-SIAM Symposium on Discrete Algorithms, pp. 205–214 (2003)

  3. Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and intractability of approximation problems. J. ACM 45(3), 501–555 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bartal, Y., Gonen, R., Nisan, N.: Incentive compatible multi unit combinatorial auctions. In: Theoretical Aspects of Rationality and Knowledge, pp. 72–87 (2003)

  5. Blumrosen, L., Nisan, N.: On the computational power of ascending auctions 1: Demand queries. In: ACM Conference on Electronic Commerce, pp. 29–43 (2005)

  6. Cramton, P., Shoham, Y., Steinberg, R. (eds.): Combinatorial Auctions. MIT Press, Cambridge (2006)

    Google Scholar 

  7. Dobzinski, S., Nisan, N., Schapira, M.: Approximation algorithms for combinatorial auctions with complement-free bidders. In: ACM Symposium on Theory of Computing, pp. 610–618 (2005)

  8. Dobzinski, S., Nisan, N., Schapira, M.: Truthful randomized mechanisms for combinatorial auctions. In: ACM Symposium on Theory of Computing, pp. 51–60 (2006)

  9. Dobzinski, S., Schapira, M.: Private communication (2006)

  10. Dobzinski, S., Schapira, M.: An improved approximation algorithm for combinatorial auctions with submodular bidders. In: ACM-SIAM Symposium on Discrete Algorithms, pp. 1064–1073 (2006)

  11. Feige, U.: A threshold of lnn for approximating set cover. J. ACM 45(4), 634–652 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  12. Feige, U.: On maximizing welfare when utility functions are subadditive. In: ACM Symposium on Theory of Computing, pp. 41–50 (2006)

  13. Feige, U., Halldorsson, M.M., Kortsarz, G., Srinivasan, A.: Approximating the domatic number. SIAM J. Comput. 32(1), 172–195 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  14. Feige, U., Kilian, J.: Zero knowledge and the chromatic number. J. Comput. Syst. Sci. 57, 187–199 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  15. Feige, U., Vondrak, J.: The allocation problem with submodular utility functions. In: IEEE Symposium on Foundations of Computer Science (2006)

  16. Gul, F., Stacchetti, E.: Walrasian equilibrium with gross substitutes. J. Econ. Theory 87, 66–95 (2000)

    Article  MathSciNet  Google Scholar 

  17. Holzman, R., Kfir-Dahav, N., Monderer, D., Tennenholtz, M.: Bundling equilibrium in combinatorial auctions. Games Econ. Behav. 47, 104–123 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  18. Khot, S., Lipton, R.J., Markakis, E., Mehta, A.: Inapproximability results for combinatorial auctions with submodular utility functions. In: Workshop on Internet and Network Economics, pp. 92–101 (2005)

  19. Lehmann, B., Lehmann, D., Nisan, N.: Combinatorial auctions with decreasing marginal utilities. In: ACM Conference on Electronic Commerce, pp. 18–28 (2001)

  20. Lehmann, D., O’Callaghan, L., Shoham, Y.: Truth revelation in approximately efficient combinatorial auctions. In: ACM Conference on Electronic Commerce, pp. 96–102 (1999)

  21. Lund, C., Yannakakis, M.: On the hardness of approximating minimization problems. J. ACM 41(5), 960–981 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  22. Nisan, N., Segal, I.: The communication requirements of efficient allocations and supporting lindahl prices. J. Econ. Theory 129(1), 192–224 (2004)

    Article  MathSciNet  Google Scholar 

  23. Petrank, E.: The hardness of approximation: Gap location. Comput. Complex. 4, 133–157 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  24. Raz, R.: A parallel repetition theorem. SIAM J. Comput. 27(3), 763–803 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  25. Sandholm, T.: An algorithm for optimal winner determination in combinatorial auctions. Artif. Intell. 135, 1–54 (2002)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Subhash Khot & Richard J. Lipton

  2. Telcordia Research, Morristown, NJ, 07960, USA

    Richard J. Lipton

  3. Dept. of Computer Science, University of Toronto, 10 King’s College Road, Toronto, ON, M5S3G4, Canada

    Evangelos Markakis

  4. IBM Almaden Research Center, 650 Harry Rd, San Jose, CA, 95120, USA

    Aranyak Mehta

Authors
  1. Subhash Khot
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  2. Richard J. Lipton
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  3. Evangelos Markakis
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  4. Aranyak Mehta
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Corresponding author

Correspondence to Evangelos Markakis.

Additional information

This work was performed when all authors were at the Georgia Institute of Technology. A preliminary version of this work appears in Khot et al. (Workshop on internet and network economics, pp. 92–101, 2005)

R.J. Lipton’s research supported by NSF grant CCF-0431023.

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Khot, S., Lipton, R.J., Markakis, E. et al. Inapproximability Results for Combinatorial Auctions with Submodular Utility Functions. Algorithmica 52, 3–18 (2008). https://doi.org/10.1007/s00453-007-9105-7

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  • DOI: https://doi.org/10.1007/s00453-007-9105-7

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