Approximation Algorithms for Online Weighted Rank Function Maximization under Matroid Constraints

  • Niv Buchbinder
  • Joseph (Seffi) Naor
  • R. Ravi
  • Mohit Singh
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7391)

Abstract

Consider the following online version of the submodular maximization problem under a matroid constraint. We are given a set of elements over which a matroid is defined. The goal is to incrementally choose a subset that remains independent in the matroid over time. At each time, a new weighted rank function of a different matroid (one per time) over the same elements is presented; the algorithm can add a few elements to the incrementally constructed set, and reaps a reward equal to the value of the new weighted rank function on the current set. The goal of the algorithm as it builds this independent set online is to maximize the sum of these (weighted rank) rewards. As in regular online analysis, we compare the rewards of our online algorithm to that of an offline optimum, namely a single independent set of the matroid that maximizes the sum of the weighted rank rewards that arrive over time. This problem is a natural extension of two well-studied streams of earlier work: the first is on online set cover algorithms (in particular for the max coverage version) while the second is on approximately maximizing submodular functions under a matroid constraint.

In this paper, we present the first randomized online algorithms for this problem with poly-logarithmic competitive ratio. To do this, we employ the LP formulation of a scaled reward version of the problem. Then we extend a weighted-majority type update rule along with uncrossing properties of tight sets in the matroid polytope to find an approximately optimal fractional LP solution. We use the fractional solution values as probabilities for a online randomized rounding algorithm. To show that our rounding produces a sufficiently large reward independent set, we prove and use new covering properties for randomly rounded fractional solutions in the matroid polytope that may be of independent interest.

Keywords

Competitive Ratio Rank Function Online Algorithm Combinatorial Auction 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.
    Ageev, A.A., Sviridenko, M.: Pipage rounding: A new method of constructing algorithms with proven performance guarantee. J. Comb. Optim. 8(3), 307–328 (2004)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Awerbuch, B., Azar, Y., Fiat, A., Leighton, T.: Making commitments in the face of uncertainty: how to pick a winner almost every time (extended abstract). In: STOC 1996: Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, pp. 519–530 (1996)Google Scholar
  3. 3.
    Babaioff, M., Immorlica, N., Kleinberg, R.: Matroids, secretary problems, and online mechanisms. In: ACM-SIAM Symposium on Discrete Algorithms, pp. 434–443 (2007)Google Scholar
  4. 4.
    Bateni, M., Hajiaghayi, M., Zadimoghaddam, M.: Submodular Secretary Problem and Extensions. In: Serna, M., Shaltiel, R., Jansen, K., Rolim, J. (eds.) APPROX and RANDOM 2010. LNCS, vol. 6302, pp. 39–52. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  5. 5.
    Borodin, A., El-Yaniv, R.: Online computation and competitive analysis. Cambridge University Press (1998)Google Scholar
  6. 6.
    Buchbinder, N., Naor, J.: The design of competitive online algorithms via a primal-dual approach. Foundations and Trends in Theoretical Computer Science 3(2-3), 93–263 (2009)MathSciNetGoogle Scholar
  7. 7.
    Buchbinder, N., Naor, J. (Seffi)., Ravi, R., Singh, M.: Approximation Algorithms for Online Weighted Rank Function Maximization under Matroid Constraints (2012), http://arxiv.org/abs/1205.1477
  8. 8.
    Calinescu, G., Chekuri, C., Pál, M., Vondrák, J.: Maximizing a Submodular Set Function Subject to a Matroid Constraint (Extended Abstract). In: Fischetti, M., Williamson, D.P. (eds.) IPCO 2007. LNCS, vol. 4513, pp. 182–196. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  9. 9.
    Chawla, S., Hartline, J.D., Malec, D.L., Sivan, B.: Multi-parameter mechanism design and sequential posted pricing. In: ACM Symposium on Theory of Computing, pp. 311–320 (2010)Google Scholar
  10. 10.
    Cornuejols, G., Fisher, M.L., Nemhauser, G.L.: Location of bank accounts to optimize float: An analytic study of exact and approximate algorithms. Management Science 23(8), 789–810 (1977)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Dughmi, S., Roughgarden, T., Yan, Q.: From convex optimization to randomized mechanisms: toward optimal combinatorial auctions. In: ACM Symposium on Theory of Computing, pp. 149–158 (2011)Google Scholar
  12. 12.
    Edmonds, J.: Submodular functions, matroids, and certain polyhedra. In: Proceedings of the Calgary International Conference on Combinatorial Structures and their Application, pp. 69–87. Gordon and Breach, New York (1969)Google Scholar
  13. 13.
    Fisher, M.L., Nemhauser, G.L., Wolsey, L.A.: An analysis of approximations for maximizing submodular set functions - part ii. Mathematical Programming 14, 265–294 (1978)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Goundan, P.R., Schulz, A.S.: Revisiting the greedy approach to submodular set function maximization (January 2009) (preprint)Google Scholar
  15. 15.
    Karger, D.R.: Random sampling and greedy sparsification for matroid optimization problems. Mathematical Programming 82, 99–116 (1998)MathSciNetGoogle Scholar
  16. 16.
    Khot, S., Lipton, R.J., Markakis, E., Mehta, A.: Inapproximability results for combinatorial auctions with submodular utility functions. Algorithmica 52(1), 3–18 (2008)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Lehmann, B., Lehmann, D.J., Nisan, N.: Combinatorial auctions with decreasing marginal utilities. In: ACM Conference on Electronic Commerce, pp. 18–28 (2001)Google Scholar
  18. 18.
    Mirrokni, V.S., Schapira, M., Vondrák, J.: Tight information-theoretic lower bounds for welfare maximization in combinatorial auctions. In: ACM Conference on Electronic Commerce, pp. 70–77 (2008)Google Scholar
  19. 19.
    Nemhauser, G.L., Wolsey, L.A.: Best Algorithms for Approximating the Maximum of a Submodular Set Function. Mathematics of Operations Research 3(3), 177–188 (1978)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Schrijver, A.: Combinatorial optimization - polyhedra and efficiency. Springer (2005)Google Scholar
  21. 21.
    Vondrak, J.: Optimal approximation for the submodular welfare problem in the value oracle model. In: STOC 2008: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, pp. 67–74 (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Niv Buchbinder
    • 1
  • Joseph (Seffi) Naor
    • 2
  • R. Ravi
    • 3
  • Mohit Singh
    • 4
    • 5
  1. 1.Computer Science Dept.Open University of IsraelIsrael
  2. 2.Computer Science Dept.TechnionHaifaIsrael
  3. 3.Tepper School of BusinessCarnegie Mellon UniversityPittsburghUSA
  4. 4.Microsoft ResearchRedmondUSA
  5. 5.School of Computer ScienceMcGill UniversityMontrealCanada

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