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Efficient Submodular Function Maximization under Linear Packing Constraints

  • Yossi Azar
  • Iftah Gamzu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7391)

Abstract

We study the problem of maximizing a monotone submodular set function subject to linear packing constraints. An instance of this problem consists of a matrix A ∈ [0,1] m ×n , a vector b ∈ [1, ∞ ) m , and a monotone submodular set function f: 2[n] → ℝ + . The objective is to find a set S that maximizes f(S) subject to A x S  ≤ b, where x S stands for the characteristic vector of the set S. A well-studied special case of this problem is when f is linear. This special linear case captures the class of packing integer programs.

Our main contribution is an efficient combinatorial algorithm that achieves an approximation ratio of Ω(1 / m 1/W ), where W =  min {b i / A ij : A ij  > 0} is the width of the packing constraints. This result matches the best known performance guarantee for the linear case. One immediate corollary of this result is that the algorithm under consideration achieves constant factor approximation when the number of constraints is constant or when the width of the constraints is sufficiently large. This motivates us to study the large width setting, trying to determine its exact approximability. We develop an algorithm that has an approximation ratio of (1 − ε)(1 − 1/e) when W = Ω(ln m / ε 2). This result essentially matches the theoretical lower bound of 1 − 1/e. We also study the special setting in which the matrix A is binary and k-column sparse. A k-column sparse matrix has at most k non-zero entries in each of its column. We design a fast combinatorial algorithm that achieves an approximation ratio of Ω(1 / (Wk 1/W )), that is, its performance guarantee only depends on the sparsity and width parameters.

Keywords

Approximation Ratio Combinatorial Algorithm Cardinality Constraint Submodular Function Approximation Guarantee 
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)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Azar, Y., Regev, O.: Combinatorial algorithms for the unsplittable flow problem. Algorithmica 44(1), 49–66 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bansal, N., Korula, N., Nagarajan, V., Srinivasan, A.: On k-Column Sparse Packing Programs. In: Eisenbrand, F., Shepherd, F.B. (eds.) IPCO 2010. LNCS, vol. 6080, pp. 369–382. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  4. 4.
    Briest, P., Krysta, P., Vöcking, B.: Approximation techniques for utilitarian mechanism design. In: 37th STOC, pp. 39–48 (2005)Google Scholar
  5. 5.
    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
  6. 6.
    Chekuri, C., Khanna, S.: On multidimensional packing problems. SIAM J. Comput. 33(4), 837–851 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Chekuri, C., Vondrák, J.: Personal Communication (2010)Google Scholar
  8. 8.
    Chekuri, C., Vondrák, J., Zenklusen, R.: Dependent randomized rounding via exchange properties of combinatorial structures. In: 51st FOCS, pp. 575–584 (2010)Google Scholar
  9. 9.
    Chekuri, C., Vondrák, J., Zenklusen, R.: Submodular function maximization via the multilinear relaxation and contention resolution schemes. In: 43rd STOC, pp. 783–792 (2011)Google Scholar
  10. 10.
    Dobzinski, S., Vondrák, J.: From query complexity to computational complexity. In: 44th STOC (2012)Google Scholar
  11. 11.
    Feige, U.: A threshold of ln n for approximating set cover. J. ACM 45(4), 634–652 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Feige, U., Mirrokni, V.S., Vondrák, J.: Maximizing non-monotone submodular functions. In: 48th FOCS, pp. 461–471 (2007)Google Scholar
  13. 13.
    Feldman, M., Naor, J(S.), Schwartz, R.: Nonmonotone Submodular Maximization via a Structural Continuous Greedy Algorithm (Extended Abstract). In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011 Part I. LNCS, vol. 6755, pp. 342–353. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  14. 14.
    Feldman, M., Naor, J., Schwartz, R.: A unified continuous greedy algorithm for submodular maximization. In: 52nd FOCS, pp. 570–579 (2011)Google Scholar
  15. 15.
    Fisher, M.L., Nemhauser, G.L., Wolsey, L.A.: An analysis of approximations for maximizing submodular set functions ii. Math. Program. Study 8, 73–87 (1978)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Garg, N., Könemann, J.: Faster and simpler algorithms for multicommodity flow and other fractional packing problems. SIAM J. Comput. 37(2), 630–652 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Gharan, S.O., Vondrák, J.: Submodular maximization by simulated annealing. In: 22nd SODA, pp. 1098–1116 (2011)Google Scholar
  18. 18.
    Goundan, P.R., Schulz, A.S.: Revisiting the greedy approach to submodular set function maximization (2007) (manuscript)Google Scholar
  19. 19.
    Khuller, S., Moss, A., Naor, J.: The budgeted maximum coverage problem. Inf. Process. Lett. 70(1), 39–45 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Kulik, A., Shachnai, H., Tamir, T.: Maximizing submodular set functions subject to multiple linear constraints. In: 20th SODA, pp. 545–554 (2009)Google Scholar
  21. 21.
    Lee, J., Mirrokni, V.S., Nagarajan, V., Sviridenko, M.: Maximizing nonmonotone submodular functions under matroid or knapsack constraints. SIAM J. Discrete Math. 23(4), 2053–2078 (2010)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Nemhauser, G.L., Wolsey, L.A.: Best algorithms for approximating the maximum of a submodular set function. Math. Operations Research 3(3), 177–188 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Nemhauser, G.L., Wolsey, L.A., Fisher, M.L.: An analysis of approximations for maximizing submodular set functions i. Math. Program. 14, 265–294 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Raghavan, P.: Probabilistic construction of deterministic algorithms: Approximating packing integer programs. Journal of Computer and System Sciences 37(2), 130–143 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Raghavan, P., Thompson, C.D.: Randomized rounding: a technique for provably good algorithms and algorithmic proofs. Combinatorica 7(4), 365–374 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Srinivasan, A.: Improved approximation guarantees for packing and covering integer programs. SIAM J. Comput. 29(2), 648–670 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Sviridenko, M.: A note on maximizing a submodular set function subject to a knapsack constraint. Oper. Res. Lett. 32(1), 41–43 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Vondrák, J.: Optimal approximation for the submodular welfare problem in the value oracle model. In: 40th STOC, pp. 67–74 (2008)Google Scholar
  29. 29.
    Vondrák, J.: Symmetry and approximability of submodular maximization problems. In: 50th FOCS, pp. 651–670 (2009)Google Scholar
  30. 30.
    Wolsey, L.A.: Maximising real-valued submodular functions: Primal and dual heuristics for location problems. Math. Operations Research 7(3), 410–425 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Young, N.E.: Randomized rounding without solving the linear program. In: 6th SODA, pp. 170–178 (1995)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Yossi Azar
    • 1
  • Iftah Gamzu
    • 1
    • 2
  1. 1.Blavatnik School of Computer ScienceTel-Aviv Univ.Israel
  2. 2.Computer Science DivisionThe Open Univ.Israel

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