Maximizing a Submodular Set Function Subject to a Matroid Constraint (Extended Abstract)

  • Gruia Calinescu
  • Chandra Chekuri
  • Martin Pál
  • Jan Vondrák
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4513)


Let \(f:2^{N} \rightarrow \cal R^{+}\) be a non-decreasing submodular set function, and let \((N,\cal I)\) be a matroid. We consider the problem \(\max_{S \in \cal I} f(S)\). It is known that the greedy algorithm yields a 1/2-approximation [9] for this problem. It is also known, via a reduction from the max-k-cover problem, that there is no (1 − 1/e + ε)-approximation for any constant ε> 0, unless P = NP [6]. In this paper, we improve the 1/2-approximation to a (1 − 1/e)-approximation, when f is a sum of weighted rank functions of matroids. This class of functions captures a number of interesting problems including set coverage type problems. Our main tools are the pipage rounding technique of Ageev and Sviridenko [1] and a probabilistic lemma on monotone submodular functions that might be of independent interest.

We show that the generalized assignment problem (GAP) is a special case of our problem; although the reduction requires |N| to be exponential in the original problem size, we are able to interpret the recent (1 − 1/e)-approximation for GAP by Fleischer et al. [10] in our framework. This enables us to obtain a (1 − 1/e)-approximation for variants of GAP with more complex constraints.


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  1. 1.
    Ageev, A., Sviridenko, M.: Pipage rounding: a new method of constructing algorithms with proven performance guarantee. J. of Combinatorial Optimization 8, 307–328 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Chekuri, C., Kumar, A.: Maximum coverage problem with group budget constraints and applications. In: Jansen, K., Khanna, S., Rolim, J.D.P., Ron, D. (eds.) RANDOM 2004 and APPROX 2004. LNCS, vol. 3122, pp. 72–83. Springer, Heidelberg (2004)Google Scholar
  3. 3.
    Chekuri, C., Pál, M.: A recursive greedy algorithm for walks in directed graphs. In: Proc. of IEEE FOCS (2005)Google Scholar
  4. 4.
    Chekuri, C., Khanna, S.: A PTAS for the multiple knapsack problem. SIAM J. on Computing 35(3), 713–728 (2004)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Cornuejols, G., Fisher, M., Nemhauser, G.: Location of bank accounts to optimize float: an analytic study of exact and approximate algorithms. Management Science 23, 789–810 (1977)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Feige, U.: A threshold of ln n for approximating set cover. JACM 45(4), 634–652 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Feige, U.: On maximizing welfare when utility functions are subadditive. In: Proc. of ACM STOC, pp. 41–50 (2006)Google Scholar
  8. 8.
    Feige, U., Vondrák, J.: Approximation algorithms for allocation problems: Improving the Factor of 1 − 1/e. In: Proc. of IEEE FOCS, pp. 667–676 (2006)Google Scholar
  9. 9.
    Fisher, M.L., Nemhauser, G.L., Wolsey, L.A.: An analysis of approximations for maximizing submodular set functions - II. Math. Prog. Study 8, 73–87 (1978)MathSciNetGoogle Scholar
  10. 10.
    Fleischer, L., Goemans, M.X., Mirrokni, V.S., Sviridenko, M.: Tight approximation algorithms for maximum general assignment problems. In: Proc. of ACM-SIAM SODA, pp. 611–620 (2006)Google Scholar
  11. 11.
    Hazan, E., Safra, S., Schwartz, O.: On the complexity of approximating k-set packing. In: Proc. of APPROX (2003)Google Scholar
  12. 12.
    Jenkyns, T.A.: The efficiency of the “greedy” algorithm. In: Proc. of 7th South Eastern Conference on Combinatorics, Graph Theory and Computing, pp. 341–350 (1976)Google Scholar
  13. 13.
    Korte, B., Hausmann, D.: An analysis of the greedy heuristic for independence systems. Annals of Discrete Math, 2, 65–74 (1978)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Nemhauser, G.L., Wolsey, L.A., Fisher, M.L.: An analysis of approximations for maximizing submodular set functions - I. Math. Prog. 14, 265–294 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Gandhi, R., Khuller, S., Parthasarathy, S., Srinivasan, A.: Dependent rounding and its applications to approximation algorithms. JACM 53(3), 324–360 (2006)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Schrijver, A.: Combinatorial optimization - polyhedra and efficiency. Springer, Heidelberg (2003)zbMATHGoogle Scholar
  17. 17.
    Srinivasan, A.: Distributions on level-sets with applications to approximation algorithms. In: Proc. of IEEE FOCS, pp. 588–597 (2001)Google Scholar
  18. 18.
    Wolsey, L.: An analysis of the greedy algorithm for the submodular set covering problem. Combinatorica 2, 385–393 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Wolsey, L.: Maximizing real-valued submodular functions: Primal and dual heuristics for location Problems. Math. of Operations Research 7, 410–425 (1982)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Gruia Calinescu
    • 1
  • Chandra Chekuri
    • 2
  • Martin Pál
    • 3
  • Jan Vondrák
    • 4
  1. 1.Computer Science Dept., Illinois Institute of Technology, Chicago, IL 
  2. 2.Dept. of Computer Science, University of Illinois, Urbana, IL 61801 
  3. 3.Google Inc., 1440 Broadway, New York, NY 10018 
  4. 4.Dept. of Mathematics, Princeton University, Princeton, NJ 08544 

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