Skip to main content
SpringerLink
Log in

The Simulated Greedy Algorithm for Several Submodular Matroid Secretary Problems

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

We study the matroid secretary problems with submodular valuation functions. In these problems, the elements arrive in random order. When one element arrives, we have to make an immediate and irrevocable decision on whether to accept it or not. The set of accepted elements must form an independent set in a predefined matroid. Our objective is to maximize the value of the accepted elements. In this paper, we focus on the case that the valuation function is a non-negative and monotonically non-decreasing submodular function. We introduce a general algorithm for such submodular matroid secretary problems. In particular, we obtain constant competitive algorithms for the cases of laminar matroids and transversal matroids. Our algorithms can be further applied to any independent set system defined by the intersection of a constant number of laminar matroids, while still achieving constant competitive ratios. Notice that laminar matroids generalize uniform matroids and partition matroids. On the other hand, when the underlying valuation function is linear, our algorithm achieves a competitive ratio of 9.6 for laminar matroids, which significantly improves the previous result.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1 We used \(\mathcal {F}\) to denote a laminar matroid instead of \(\mathcal {I}\) as in Section 6

  2. 2 We define g(e) based on N e instead of S e , i.e., the current set of elements in S, because S e is still a random set even all the randomness before e’s arrival is fixed.

References

  1. Babaioff, M., Dinitz, M., Gupta, A., Immorlica, N., Talwar, K.: Secretary problems: weights and discounts. In: SODA, pp 1245–1254 (2009)

  2. Bateni, M.H., Hajiaghayi, M.T., Zadimoghaddam, M.: Submodular secretary problem and extensions. ACM Trans. Algorithms 9(4), 32 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Babaioff, M., Immorlica, N., Kleinberg, R.: Matroids, secretary problems, and online mechanisms. In: SODA, pp 434–443 (2007)

  4. Babaioff, M., Immorlica, N., Kempe, D., Kleinberg, R.: A knapsack secretary problem with applications. In: APPROX-RANDOM, pp 16–28 (2007)

  5. Chakraborty, S., Lachish, O.: Improved competitive ratio for the matroid secretary problem. In: SODA, pp 1702–1712 (2012)

  6. Dinitz, M., Kortsarz, G.: Matroid secretary for regular and decomposable matroids. SIAM J. Comput. 43(5), 1807–1830 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dimitrov, N.B., Plaxton, C.G.: Competitive weighted matching in transversal matroids. Algorithmica 62(1-2), 333–348 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dynkin, E.B.: Optimal choice of the stopping moment of a markov process. Dokl. Akad. Nauk SSSR 150, 238–240 (1963)

    MathSciNet  Google Scholar 

  9. Ferguson, T.S.: Who solved the secretary problem Stat. Sci. 4, 282–289 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  10. Feldman, M., Naor, J. (Seffi), Schwartz, R.: Improved competitive ratios for submodular secretary problems (extended abstract). In: APPROX-RANDOM, pp 218–229 (2011)

  11. Fisher, M. L., Nemhauser, G. L., Wolsey, L. A.: An analysis of approximations for maximizing submodular set functions — II. In: Polyhedral combinatorics of mathematical programming studies, vol. 8, pp 73–87. Springer, Berlin Heidelberg (1978)

  12. Freeman, P R: The secretary problem and its extensions: A review, International Statistical Review (1983)

  13. Moran Feldman, Ola Svensson, Rico Zenklusen: A simple O(log log(rank))-competitive algorithm for the matroid secretary problem. In: Piotr Indyk (ed.) Proceedings of the twenty-sixth annual ACM-SIAM symposium on discrete algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pp 1189–1201. SIAM (2015)

  14. Gardner, M.: Mathematical games column. Sci. Am., 35 (1960)

  15. Gupta, A., Roth, A., Schoenebeck, G., Talwar, K.: Constrained non-monotone submodular maximization: offline and secretary algorithms. In: WINE, pp 246–257 (2010)

  16. Gharan, S. O., Vondrȧk, J.: On variants of the matroid secretary problem. Algorithmica 67(4), 472–497 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Im, S., Wang, Y.: Secretary problems: Laminar matroid and interval scheduling. In: Randall, D. (ed.) Proceedings of the twenty-second annual ACM-SIAM symposium on discrete algorithms, SODA 2011, San Francisco, California, USA, January 23-25, 2011, pp 1265–1274. SIAM (2011)

  18. Kleinberg, R.: A multiple-choice secretary algorithm with applications to online auctions. In: SODA, pp 630–631, Philadelphia (2005)

  19. Korula, N., Pál, M.: Algorithms for secretary problems on graphs and hypergraphs. In: ICALP (2), pp 508–520 (2009)

  20. Lachish, O.: O(log log rank) competitive-ratio for the matroid secretary problem. arXiv:1403.7343 (2014)

  21. Soto, J.A.: Matroid secretary problem in the random-assignment model. SIAM J. Comput. 42(1), 178–211 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Princeton University, Princeton, NJ, 08540, USA

    Tengyu Ma

  2. University of Liverpool, Liverpool, L69 3BX, UK

    Bo Tang

  3. Microsoft Cooperation, Sunnyvale, CA, 94089, USA

    Yajun Wang

Authors
  1. Tengyu Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bo Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yajun Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tengyu Ma.

Additional information

Part of the work was done when the authors were with Microsoft Research Asia.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ma, T., Tang, B. & Wang, Y. The Simulated Greedy Algorithm for Several Submodular Matroid Secretary Problems. Theory Comput Syst 58, 681–706 (2016). https://doi.org/10.1007/s00224-015-9642-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-015-9642-4

Keywords

Search

Navigation