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Opportunity Cost Algorithms for Combinatorial Auctions

  • Karhan Akcoglu
  • James Aspnes
  • Bhaskar DasGupta
  • Ming-Yang Kao
Part of the Applied Optimization book series (APOP, volume 74)

Abstract

Two general algorithms based on opportunity costs are given for approximating a revenue-maximizing set of bids an auctioneer should accept, in a combinatorial auction in which each bidder offers a price for some subset of the available goods and the auctioneer can only accept non-intersecting bids. Since this problem is difficult even to approximate in general, the algorithms are most useful when the bids are restricted to be connected node subsets of an underlying object graph that represents which objects are relevant to each other. The approximation ratios of the algorithms depend on structural properties of this graph and are small constants for many interesting families of object graphs. The running times of the algorithms are linear in the size of the bid graph, which describes the conflicts between bids. Extensions of the algorithms allow for efficient processing of additional constraints, such as budget constraints that associate bids with particular bidders and limit how many bids from a particular bidder can be accepted.

Keywords

combinatorial auction winner determination budget constraints object graphs bid graphs graph connectivity computational hardness approximation algorithms opportunity costs 

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References

  1. [Mon et al., 1995]
    Alon, N., Feige, U., Wigderson, A., and Zuckerman, D. (1995). Derandomized graph products. Computational Complexity, 5(1): 6075.CrossRefGoogle Scholar
  2. [Bafna et al., 1999]
    Bafna, V., Berman, P., and Fujito, T. (1999). A 2-approximation algorithm for the undirected feedback vertex set problem. SIAM Journal on Discrete Mathematics, 12: 289–297.CrossRefGoogle Scholar
  3. [Banks et al., 1989]
    Banks, J. S., Ledyard, J. 0., and Porter, D. (1989). Allocating uncertain and unresponsive resources: An experimental approach. The Rand Journal of Economics, 20: 1–25.Google Scholar
  4. [Bar-Noy et al., 2000]
    Bar-Noy, A., Bar-Yehuda, R., Freund, A., Naor, J. S., and Schieber, B. (2000). A unified approach to approximating resource allocation and scheduling. In Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, pages 735–744.Google Scholar
  5. [Bar-Yehuda, 2000]
    Bar-Yehuda, R. (2000). One for the price of two• a unified approach for approximating covering problems. Algorithmica,27(2):131144.Google Scholar
  6. [Bar-Yehuda and Even, 1985]
    Bar-Yehuda, R. and Even, S. (1985). A local-ratio theorem for approximating the weighted vertex cover problem. Annals of Discrete Mathematics, 25: 27–45.Google Scholar
  7. [Berman and DasGupta, 2000]
    Berman, P. and DasGupta, B. (2000). Improvements in throughput maximization for real-time scheduling. In Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, pages 680687.Google Scholar
  8. [Chen et al., 2000]
    Chen, Y., Kao, M. Y., and Lu, H. I. (2000). Optimal bid sequences for multiple-object auctions with unequal budgets. In Lee, D. T. and Teng, S. H., editors, Lecture Notes in Computer Science 1969: Proceedings of the 11th Annual International Symposium on Algorithms and Computation, pages 84–95. Springer-Verlag, New York, NY.Google Scholar
  9. [Clearwater, 1996]
    Clearwater, S. H., editor (1996). Market-Based Control, a Paradigm for Distributed Resource Allocation. World Scientific, River Ridge, NJ.Google Scholar
  10. [de Vries and Vohra, 2000]
    de Vries, S. and Vohra, R. (2000). Combinatorial auctions: A survey. Manuscript.Google Scholar
  11. [DeMartini et al., 1999]
    DeMartini, C., Kwasnica, A. M., Ledyard, J. O., and Porter, D. (1999). A new and improved design for multiple-object iterative auctions. Technical Report SSWP 1054, California Institute of Technology.Google Scholar
  12. [Fujishima et al., 1999]
    Fujishima, Y., Leyton-Brown, K., and Shoham, Y. (1999). Taming the computational complexity of combinatorial auctions: Optimal and approximate approaches. In Proceedings of the 16th International Joint Conference on Artificial Intelligence (IJCAI), pages 548–553, Stockholm, Sweden.Google Scholar
  13. [Gale, 1990]
    Gale, I. (1990). A multiple-object auction with superadditive values. Economics Letters, 34 (4): 323–328.CrossRefGoogle Scholar
  14. [Gavril, 1972]
    Gavril, F. (1972). Algorithms for minimum coloring, maximum clique, minimum covering by cliques, and maximum independent set of a chordal graph. SIAM Journal on Computing, 1 (2): 180–187.CrossRefGoogle Scholar
  15. [Grötschel et al., 1988]
    Grötschel, M., Lovâsz, L., and Schrijver, A. (1988). Geometric Algorithms and Combinatorial Optimization, volume 2 of Algorithms and Combinatorics. Springer-Verlag, New York, NY.Google Scholar
  16. [Halldórsson and Radhakrishnan, 1994]
    Halldórsson, M. M. and Radhakrishnan, J. (1994). Improved approximations of independent sets in bounded-degree graphs via subgraph removal. Nordic Journal of Computing, 1 (4): 475–482.Google Scholar
  17. [Hâstad, 1999]
    Hâstad, J. (1999). Clique is hard to approximate within ni - E Acta Mathematica, 182 (1): 105–142.CrossRefGoogle Scholar
  18. [Hausch, 1986]
    Hausch, D. B. (1986). Multi-object auctions: sequential vs. simultaneous sales. Management Science. Journal of the Institute of Management Science. Application and Theory Series,32(12):1599–1610,1611–1612. With a comment by Michael H. Rothkopf, Elmer Dougherty and Marshall Rose.Google Scholar
  19. [Hendricks and Paarsh, 1995]
    Hendricks, K. and Paarsh, H. J. (1995). A survey of recent empirical work concerning auctions. Canadian Journal of Economics, 28 (2): 403–426.CrossRefGoogle Scholar
  20. [Hsu and Ma, 1999]
    Hsu, W.-L. and Ma, T.-H. (1999). Fast and simple algorithms for recognizing chordal comparability graphs and interval graphs. SIAM Journal on Computing,28(3):1004–1020 (electronic).Google Scholar
  21. [Kao et al., 1999]
    Kao, M. Y., Qi, J. F., and Tan, L. (1999). Optimal bidding algorithms against cheating in multiple object auctions. SIAM Journal on Computing, 28 (3): 955–969.CrossRefGoogle Scholar
  22. [Krishna and Rosenthal, 1996]
    Krishna, V. and Rosenthal, R. W. (1996). Simultaneous auctions with synergies. Games and Economic Behavior, 17 (1): 1–31.CrossRefGoogle Scholar
  23. [Lehmann et al., 1999]
    Lehmann, D., O’Callaghan, L. I., and Shoham, Y. (1999). Truth revelation in rapid approximately efficient combinatorial auctions. In Proceedings of the 1st ACM Conference on Electronic Commerce, pages 96–102. SlGecom, ACM Press.CrossRefGoogle Scholar
  24. [Lipton and Tarjan, 1979]
    Lipton, R. J. and Tarjan, R. E. (1979). A separator theorem for planar graphs. SIAM Journal on Applied Mathematics, 36 (2): 177–189.CrossRefGoogle Scholar
  25. [McAfee and McMillan, 1996]
    McAfee, R. P. and McMillan, J. (1996). Analyzing the airwaves auction. Journal of Economic Perspectives,10(1):159175.Google Scholar
  26. [McMillan and McAfee, 1987]
    McMillan, J. and McAfee, R. P. (1987). Auctions and bidding. Journal of Economic Literature, 25: 699–738.Google Scholar
  27. [Milgrom and Weber, 1982]
    Milgrom, P. R. and Weber, R. J. (1982). A theory of auctions and competitive bidding. Econometrica, 50 (5): 1089–1122.CrossRefGoogle Scholar
  28. [Palfrey, 1980]
    Palfrey, T. R. (1980). Multiple-object, discriminatory auctions with bidding constraints: a game-theoretic analysis. Management Science. Journal of the Institute of Management Science. Application and Theory Series, 26 (9): 935–945.CrossRefGoogle Scholar
  29. [Parkes and Ungar, 2000]
    Parkes, D. C. and Ungar, L. H. (2000). Iterative combinatorial auctions: Theory and practice. In Proceedings of the 17th National Conference on Artificial Intelligence, pages 74–81, Austin, TX.Google Scholar
  30. [Rassenti et al., 1982]
    Rassenti, S. J., Smith, V. L., and Bulfin, R. L. (1982). A combinatorial mechanism for airport time slot allocation. Bell Journal of Economics, 13: 402–417.CrossRefGoogle Scholar
  31. [Reed, 1992]
    Reed, B. A. (1992). Finding approximate separators and computing tree width quickly. In Proceedings of the 24th Annual ACM Symposium on Theory of Computing, pages 221–228.Google Scholar
  32. [Robertson and Seymour, 1986]
    Robertson, N. and Seymour, P. D. (1986). Graph minors. II. Algorithmic aspects of tree-width. Journal of Algorithms, 7 (3): 309–322.CrossRefGoogle Scholar
  33. [Rose et al., 1976]
    Rose, D. J., Tarjan, R. E., and Lueker, G. S. (1976). Algorithmic aspects of vertex elimination on graphs. SIAM Journal on Computing, 5 (2): 266–283.CrossRefGoogle Scholar
  34. [Rothkopf et al., 1998]
    Rothkopf, M. H., Pekec, A., and Harstad, R. M. (1998). Computationally manageable combinatorial auctions. Management Science, 44 (8): 1131–1147.CrossRefGoogle Scholar
  35. [Sandholm and Suri, 2000]
    Sandholm, T. W. and Suri, S. (2000). Improved algorithms for optimal winner determination in combinatorial auctions and generalizations. In Proceedings of the 17th National Conference onArtificial Intelligence, pages 90–97, Austin, TX.Google Scholar
  36. [Wilson, 1992]
    Wilson, R. (1992). Strategic analysis of auctions. In Aumann, R. J. and Hart, S., editors, Handbook of Game Theory with Economic Applications, volume 1, pages 227–279. Elsevier Science, New York, NY.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2002

Authors and Affiliations

  • Karhan Akcoglu
    • 1
  • James Aspnes
    • 1
  • Bhaskar DasGupta
    • 2
  • Ming-Yang Kao
    • 3
  1. 1.Department of Computer ScienceYale UniversityNew HavenUSA
  2. 2.Department of Computer ScienceUniversity of Illinois at ChicagoChicagoUSA
  3. 3.Department of Computer ScienceNorthwestern UniversityEvanstonUSA

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