Secretary Problems with Convex Costs

  • Siddharth Barman
  • Seeun Umboh
  • Shuchi Chawla
  • David Malec
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7391)

Abstract

We consider online resource allocation problems where given a set of requests our goal is to select a subset that maximizes a value minus cost type of objective. Requests are presented online in random order, and each request possesses an adversarial value and an adversarial size. The online algorithm must make an irrevocable accept/reject decision as soon as it sees each request. The “profit” of a set of accepted requests is its total value minus a convex cost function of its total size. This problem falls within the framework of secretary problems. Unlike previous work in that area, one of the main challenges we face is that the objective function can be positive or negative, and we must guard against accepting requests that look good early on but cause the solution to have an arbitrarily large cost as more requests are accepted. This necessitates new techniques. We study this problem under various feasibility constraints and present online algorithms with competitive ratios only a constant factor worse than those known in the absence of costs for the same feasibility constraints. We also consider a multi-dimensional version of the problem that generalizes multi-dimensional knapsack within a secretary framework. In the absence of feasibility constraints, we present an O(ℓ) competitive algorithm where ℓ is the number of dimensions; this matches within constant factors the best known ratio for multi-dimensional knapsack secretary.

Keywords

Competitive Ratio Online Algorithm Full Version Competitive Algorithm Submodular Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Babaioff, M., Dinitz, M., Gupta, A., Immorlica, N., Talwar, K.: Secretary problems: weights and discounts. In: SODA 2009 (2009)Google Scholar
  2. 2.
    Babaioff, M., Hartline, J., Kleinberg, R.: Selling banner ads: Online algorithms with buyback. In: Fourth Workshop on Ad Auctions (2008)Google Scholar
  3. 3.
    Babaioff, M., Hartline, J.D., Kleinberg, R.D.: Selling ad campaigns: Online algorithms with cancellations. In: EC 2009 (2009)Google Scholar
  4. 4.
    Babaioff, M., Immorlica, N., Kempe, D., Kleinberg, R.: A knapsack secretary problem with applications. In: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pp. 16–28 (2007)Google Scholar
  5. 5.
    Babaioff, M., Immorlica, N., Kleinberg, R.: Matroids, secretary problems, and online mechanisms. In: SODA 2007 (2007)Google Scholar
  6. 6.
    Barman, S., Umboh, S., Chawla, S., Malec, D.L.: Secretary problems with convex costs. CoRR, abs/1112.1136 (2011)Google Scholar
  7. 7.
    Bateni, M.H., Hajiaghayi, M.T., Zadimoghaddam, M.: Submodular secretary problem and extensions. In: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pp. 39–52 (2010)Google Scholar
  8. 8.
    Chakraborty, S., Lachish, O.: Improved competitive ratio for the matroid secretary problem. In: SODA 2012 (2012)Google Scholar
  9. 9.
    Chawla, S., Hartline, J.D., Malec, D.L., Sivan, B.: Multi-parameter mechanism design and sequential posted pricing. In: STOC 2010 (2010)Google Scholar
  10. 10.
    Dean, B.C., Goemans, M.X., Vondrák, J.: Adaptivity and approximation for stochastic packing problems. In: SODA 2005 (2005)Google Scholar
  11. 11.
    Dimitrov, N.B., Plaxton, C.G.: Competitive Weighted Matching in Transversal Matroids. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 397–408. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  12. 12.
    Dynkin, E.B.: The optimum choice of the instant for stopping a Markov process. Soviet Math. Dokl 4(627-629) (1963)Google Scholar
  13. 13.
    Feige, U., Flaxman, A.D., Hartline, J.D., Kleinberg, R.D.: On the Competitive Ratio of the Random Sampling Auction. In: Deng, X., Ye, Y. (eds.) WINE 2005. LNCS, vol. 3828, pp. 878–886. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  14. 14.
    Feldman, M., Naor, J., Schwartz, R.: Improved competitive ratios for submodular secretary problems. In: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pp. 218–229 (2011)Google Scholar
  15. 15.
    Ferguson, T.: Who solved the secretary problem. Statist. Sci. 4(3), 282–289 (1989)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Freeman, P.R.: The secretary problem and its extensions: a review. International Statistical Review 51(2), 189–206 (1983)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Gupta, A., Roth, A., Schoenebeck, G., Talwar, K.: Constrained non-monotone submodular maximization: Offline and secretary algorithms. Internet and Network Economics, 246–257 (2010)Google Scholar
  18. 18.
    Hajiaghayi, M.T., Kleinberg, R., Parkes, D.C.: Adaptive limited-supply online auctions. In: EC 2004 (2004)Google Scholar
  19. 19.
    Kennedy, D.P.: Prophet-type inequalities for multi-choice optimal stopping. Stochastic Processes and their Applications 24(1), 77–88 (1987)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Kleinberg, R.: A multiple-choice secretary algorithm with applications to online auctions. In: SODA 2005 (2005)Google Scholar
  21. 21.
    Korula, N., Pál, M.: Algorithms for Secretary Problems on Graphs and Hypergraphs. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5556, pp. 508–520. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  22. 22.
    Oxley, J.: Matroid Theory. Oxford University Press (1992)Google Scholar
  23. 23.
    Samuel-Cahn, E.: Comparison of threshold stop rules and maximum for independent nonnegative random variables. The Annals of Probability 12 (1984)Google Scholar
  24. 24.
    Samuels, S.: Secretary problems. In: Handbook of Sequential Analysis, pp. 381–405. Marcel Dekker (1991)Google Scholar
  25. 25.
    Soto, J.A.: Matroid Secretary Problem in the Random Assignment Model. In: SODA 2011 (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Siddharth Barman
    • 1
  • Seeun Umboh
    • 1
  • Shuchi Chawla
    • 1
  • David Malec
    • 1
  1. 1.University of Wisconsin–MadisonUSA

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