On the Approximability of Combinatorial Exchange Problems

  • Moshe Babaioff
  • Patrick Briest
  • Piotr Krysta
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4997)

Abstract

In a combinatorial exchange the goal is to find a feasible trade between potential buyers and sellers requesting and offering bundles of indivisible goods. We investigate the approximability of several optimization objectives in this setting and show that the problems of surplus and trade volume maximization are inapproximable even with free disposal and even if each agent’s bundle is of size at most 3. In light of the negative results for surplus maximization we consider the complementary goal of social cost minimization and present tight approximation results for this scenario. Considering the more general supply chain problem, in which each agent can be a seller and buyer simultaneously, we prove that social cost minimization remains inapproximable even with bundles of size 3, yet becomes polynomial time solvable for agents trading bundles of size 1 or 2. This yields a complete characterization of the approximability of supply chain and combinatorial exchange problems based on the size of traded bundles. We finally briefly address the problem of exchanges in strategic settings.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alon, N., Moshkovitz, D., Safra, M.: Algorithmic Construction of Sets for k-Restrictions. ACM Transactions on Algorithms 2, 153–177 (2006)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Babaioff, M., Walsh, W.E.: Incentive-Compatible, Budget-Balanced, yet Highly Efficient Auctions for Supply Chain Formation. Decision Support Systems 39, 123–149 (2005)CrossRefGoogle Scholar
  3. 3.
    Briest, P., Krysta, P., Vöcking, B.: Approximation Techniques for Utilitarian Mechanism Design. In: Proc. of STOC (2005)Google Scholar
  4. 4.
    Cook, W.J., Cunningham, W.H., Pulleyblank, W.R., Schrijver, A.: Combinatorial Optimization. John Wiley, Chichester (1998)MATHGoogle Scholar
  5. 5.
    Cramton, P., Shoham, Y., Steinberg, R.: Combinatorial Auctions. MIT Press, Cambridge (2006)Google Scholar
  6. 6.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-completeness. W.H. Freeman, New York (1979)MATHGoogle Scholar
  7. 7.
    Lehmann, D., O’Callaghan, L.I., Shoham, Y.: Truth Revelation in Approximately Efficient Combinatorial Auctions. J. of the ACM 49(5), 1–26 (2002)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Myerson, R.B., Satterthwaite, M.A.: Efficient Mechanisms for Bilateral Trading. Journal of Economic Theory 29, 265–281 (1983)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Nisan, N.: The Communication Complexity of Approximate Set Packing and Covering. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 868–875. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  10. 10.
    Nisan, N., Segal, I.: The Communication Requirements of Efficient Allocations and Supporting Prices. Journal of Economic Theory (2006)Google Scholar
  11. 11.
    Parkes, D.C., Cavallo, R., Elprin, N., Juda, A., Lahaie, S., Lubin, B., Michael, L., Shneidman, J., Sultan, H.: ICE: An Iterative Combinatorial Exchange. In: Proc. of EC, pp. 249–258 (2005)Google Scholar
  12. 12.
    Parkes, D.C., Kalagnanam, J., Eso, M.: Achieving Budget-Balance with Vickrey-Based Payment Schemes in Exchanges. In: Proc. of IJCAI, pp. 1161–1168 (2001)Google Scholar
  13. 13.
    Rajagopalan, S., Vazirani, V.V.: Primal-Dual RNC Approximation Algorithms for (multi)Set (multi)Cover and Covering Integer Programs. SIAM J. of Computing 28(2), 525–540 (1998)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Roughgarden, T., Sundararajan, M.: New Trade-Offs in Cost-Sharing Mechanisms. Proc. of STOC, 79–88 (2006)Google Scholar
  15. 15.
    Sandholm, T.: Algorithm for Optimal Winner Determination in Combinatorial Auctions. Artificial Intelligence 135(1-2), 1–54 (2002)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Sandholm, T., Suri, S.: Improved Algorithms for Optimal Winner Determination in Combinatorial Auctions and Generalizations. In: Proc. of AAAI/IAAI, pp. 90–97 (2000)Google Scholar
  17. 17.
    Sandholm, T., Suri, S., Gilpin, A., Levine, D.: Winner Determination in Combinatorial Auction Generalizations. In: Proc. of AAMAS, Bologna, Italy, pp. 69–76. ACM Press, New York (2002)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Moshe Babaioff
    • 1
  • Patrick Briest
    • 2
  • Piotr Krysta
    • 2
  1. 1.Microsoft ResearchUSA
  2. 2.Dept. of Computer ScienceUniversity of LiverpoolUK

Personalised recommendations