On Variants of the Matroid Secretary Problem

  • Shayan Oveis Gharan
  • Jan Vondrák
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6942)

Abstract

We present a number of positive and negative results for variants of the matroid secretary problem. Most notably, we design a constant-factor competitive algorithm for the “random assignment” model where the weights are assigned randomly to the elements of a matroid, and then the elements arrive on-line in an adversarial order (extending a result of Soto [20]). This is under the assumption that the matroid is known in advance. If the matroid is unknown in advance, we present an O(logr logn)-approximation, and prove that a better than O(logn / loglogn) approximation is impossible. This resolves an open question posed by Babaioff et al. [3].

As a natural special case, we also consider the classical secretary problem where the number of candidates n is unknown in advance. If n is chosen by an adversary from {1,…,N}, we provide a nearly tight answer, by providing an algorithm that chooses the best candidate with probability at least 1/(H N − 1 + 1) and prove that a probability better than 1/H N cannot be achieved (where H N is the N-th harmonic number).

Keywords

Approximation Algorithm Online Auction Secretary Problem Transversal Matroids Natural Special Case 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abdel-Hamid, A.R., Bather, J.A., Trustrum, G.B.: The secretary problem with an unknown number of candidates. J. Appl. Prob. 19, 619–630 (1982)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Babaioff, M., Dinitz, M., Gupta, A., Immorlica, N., Talwar, K.: Secretary problems: weights and discounts. In: SODA 2009, pp. 1245–1254 (2009)Google Scholar
  3. 3.
    Babaioff, M., Immorlica, N., Kleinberg, R.: Matroids, secretary problems, and online mechanisms. In: SODA 2007, pp. 434–443 (2007)Google Scholar
  4. 4.
    Bateni, M.H., Hajiaghayi, M.T., Zadimoghaddam, M.: Submodular secretary problem and extensions. In: Serna, M., Shaltiel, R., Jansen, K., Rolim, J. (eds.) APPROX 2010, LNCS, vol. 6302, pp. 39–52. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  5. 5.
    Borosan, P., Shabbir, M.: A survey of secretary problem and its extensions (preprint 2009), http://paul.rutgers.edu/~mudassir/Secretary/paper.pdf
  6. 6.
    Buchbinder, N., Jain, K., Singh, M.: Secretary problems via linear programming. In: Eisenbrand, F., Shepherd, F.B. (eds.) IPCO 2010. LNCS, vol. 6080, pp. 163–176. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  7. 7.
    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
  8. 8.
    Dynkin, E.B.: The optimum choice of the instant for stopping a markov process. Soviet Mathematics, Doklady 4 (1963)Google Scholar
  9. 9.
    Feldman, J., Henzinger, M., Korula, N., Mirrokni, V.S., Stein, C.: Online stochastic packing applied to display ad allocation. In: de Berg, M., Meyer, U. (eds.) ESA 2010. LNCS, vol. 6346, pp. 182–194. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  10. 10.
    Ferguson, T.S.: Who solved the secretary problem? Statistical Science 4(3), 282–289 (1989)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Gardner, M.: Mathematical Games column. Scientific American (February 1960)Google Scholar
  12. 12.
    Gilbert, J., Mosteller, F.: Recognizing the maximum of a sequence. J. Amer. Statist. Assoc. 61, 35–73 (1966)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gupta, A., Roth, A., Schoenebeck, G., Talwar, K.: Constrained non-monotone submodular maximization: Offline and secretary algorithms. In: Saberi, A. (ed.) WINE 2010. LNCS, vol. 6484, pp. 246–257. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  14. 14.
    Hajiaghayi, M.T., Kleinberg, R., Parkes, D.: Adaptive limited-supply online auctions. In: EC 2004, pp. 71–80 (2004)Google Scholar
  15. 15.
    Hajiaghayi, M.T., Kleinberg, R., Sandholm, T.: Automated online mechanism design and prophet inequalities. In: International Conference on Artificial Intelligence 2007, pp. 58–65 (2007)Google Scholar
  16. 16.
    Im, S., Wang, Y.: Secretary problems: Laminar matroid and interval scheduling. In: SODA 2011, pp. 1265–1274 (2011)Google Scholar
  17. 17.
    Lindley, D.V.: Dynamic programming and decision theory. Journal of the Royal Statistical Society. Series C (Applied Statistics) 10(1), 39–51 (1961)Google Scholar
  18. 18.
    Kleinberg, R.: A multiple-choice secretary algorithm with applications to online auctions. In: SODA 2005, pp. 630–631 (2005)Google Scholar
  19. 19.
    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
  20. 20.
    Soto, J.A.: Matroid secretary problem in the random assignment model. In: SODA 2011, pp. 1275–1284 (2011)Google Scholar
  21. 21.
    Stewart, T.J.: The secretary problem with an unknown number of options. Operations Research 1, 130–145 (1981)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Shayan Oveis Gharan
    • 1
  • Jan Vondrák
    • 2
  1. 1.Stanford UniversityStanfordUSA
  2. 2.IBM Almaden Research CenterSan JoseUSA

Personalised recommendations