Recently there has been a surge of interest in auctions research triggered on the one hand by auctions of bandwidth and other public assets and on the other by the popularity of Internet auctions and the possibility of new auction formats enabled by e-commerce. Simultaneous auction of items is a popular auction format. We consider the problem of maximizing total revenue in the simultaneous auction of a set of items where the bidders have individual budget constraints. Each bidder is permitted to bid on all the items of his choice and specifies his budget constraint to the auctioneer, who must select bids to maximize the revenue while ensuring that no budget constraints are violated. We show that the problem of maximizing revenue is such a setting is NP-hard, and present a factor-1.62 approximation algorithm for it. We formulate the problem as an integer program and solve a linear relaxation to obtain a fractional optimal solution, which is then deterministically rounded to obtain an integer solution. We argue that the loss in revenue incurred by the rounding procedure is bounded by a factor of 1.62.


auctions winner determination approximation algorithm rounding 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J-P Benoit and V. Krishna. Multiple-object auctions with budget constrained bidders. A working paper at, April 1998.
  2. 2.
    Y. Fujishima, K. Leyton-Brown, and Y. Shoham. Taming the computational complexity of combinatorial auctions: Optimal and approximate approaches. In International Joint Conferences on Artificial Intelligence, 1999.Google Scholar
  3. 3.
    R. P. McAfee and J. McMillan. Analyzing the airwaves auctions. Journal of Economic Perspectives, 10(1):159–175, 1996.Google Scholar
  4. 4.
    J. McMillan. Selling spectrum rights. Journal of Economic Perspectives, 8(3):145–162, 1994.Google Scholar
  5. 5.
    T. R. Palfrey. Multiple object, discriminatory auctions with bidding constraints: A game theoretic analysis. Management Science, 26:935–946, 1980.zbMATHMathSciNetGoogle Scholar
  6. 6.
    M. Rothkopf. Bidding in simultaneous auctions with a constraint on exposure. Operations Research, 25:620–629, 1977.zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    M. H. Rothkopf, A. Pekec, and R. M. Harstad. Computationally manageable combinatorial auctions. Management Science, 44:1131–1147, August 1998.Google Scholar
  8. 8.
    T. Sandholm. An algorithm for optimal winner determination in combinatorial auction. Technical Report WUCS-99-01, Washington University, Department of Computer Science, January 1999.Google Scholar
  9. 9.
    D.B. Shmoys and E. Tardos. An approximation algorithm for the generalised assignment problem. Mathematical Programming A, 62:461–474, 1993.CrossRefMathSciNetGoogle Scholar
  10. 10.
    Martin Spicer. International survey of spectrum assignment for cellular and pcs. Technical report, September 1996. Federal Telecommunications Commission, available at

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Rahul Garg
    • 1
  • Vijay Kumar
    • 1
  • Vinayaka Pandit
    • 1
  1. 1.IBM India Research Lab IIT DelhiNew DelhiIndia

Personalised recommendations