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Efficient Submodular Function Maximization under Linear Packing Constraints

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Automata, Languages, and Programming (ICALP 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7391))

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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.

This research was supported in part by the Israeli Centers of Research Excellence (I-CORE) program (Center No.4/11), the Israel Science Foundation (grant No. 1404/10), and by the Google Inter-university center. Due to space limitations, some proofs are omitted from this extended abstract. We refer the reader to the full version of this paper (available online at http://arxiv.org/abs/1007.3604), in which all missing details are provided.

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References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  2. Azar, Y., Regev, O.: Combinatorial algorithms for the unsplittable flow problem. Algorithmica 44(1), 49–66 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  4. Briest, P., Krysta, P., Vöcking, B.: Approximation techniques for utilitarian mechanism design. In: 37th STOC, pp. 39–48 (2005)

    Google Scholar 

  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)

    Chapter  Google Scholar 

  6. Chekuri, C., Khanna, S.: On multidimensional packing problems. SIAM J. Comput. 33(4), 837–851 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chekuri, C., Vondrák, J.: Personal Communication (2010)

    Google Scholar 

  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. 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. Dobzinski, S., Vondrák, J.: From query complexity to computational complexity. In: 44th STOC (2012)

    Google Scholar 

  11. Feige, U.: A threshold of ln n for approximating set cover. J. ACM 45(4), 634–652 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Feige, U., Mirrokni, V.S., Vondrák, J.: Maximizing non-monotone submodular functions. In: 48th FOCS, pp. 461–471 (2007)

    Google Scholar 

  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)

    Chapter  Google Scholar 

  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. 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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gharan, S.O., Vondrák, J.: Submodular maximization by simulated annealing. In: 22nd SODA, pp. 1098–1116 (2011)

    Google Scholar 

  18. Goundan, P.R., Schulz, A.S.: Revisiting the greedy approach to submodular set function maximization (2007) (manuscript)

    Google Scholar 

  19. Khuller, S., Moss, A., Naor, J.: The budgeted maximum coverage problem. Inf. Process. Lett. 70(1), 39–45 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  24. Raghavan, P.: Probabilistic construction of deterministic algorithms: Approximating packing integer programs. Journal of Computer and System Sciences 37(2), 130–143 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  25. Raghavan, P., Thompson, C.D.: Randomized rounding: a technique for provably good algorithms and algorithmic proofs. Combinatorica 7(4), 365–374 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  26. Srinivasan, A.: Improved approximation guarantees for packing and covering integer programs. SIAM J. Comput. 29(2), 648–670 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  27. Sviridenko, M.: A note on maximizing a submodular set function subject to a knapsack constraint. Oper. Res. Lett. 32(1), 41–43 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  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. Vondrák, J.: Symmetry and approximability of submodular maximization problems. In: 50th FOCS, pp. 651–670 (2009)

    Google Scholar 

  30. Wolsey, L.A.: Maximising real-valued submodular functions: Primal and dual heuristics for location problems. Math. Operations Research 7(3), 410–425 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  31. Young, N.E.: Randomized rounding without solving the linear program. In: 6th SODA, pp. 170–178 (1995)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Blavatnik School of Computer Science, Tel-Aviv Univ., Israel

    Yossi Azar & Iftah Gamzu

  2. Computer Science Division, The Open Univ., Israel

    Iftah Gamzu

Authors
  1. Yossi Azar
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  2. Iftah Gamzu
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Editor information

Editors and Affiliations

  1. Department of Computer Science and Centre for Discrete Mathematics and its Applications, University of Warwick, Warwick, UK

    Artur Czumaj

  2. Max-Planck-Institut für Informatik, Saarbrücken, Germany

    Kurt Mehlhorn

  3. Computer Laboratory, University of Cambridge, UK

    Andrew Pitts

  4. ETH Zurich, Switzerland

    Roger Wattenhofer

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Azar, Y., Gamzu, I. (2012). Efficient Submodular Function Maximization under Linear Packing Constraints. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds) Automata, Languages, and Programming. ICALP 2012. Lecture Notes in Computer Science, vol 7391. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31594-7_4

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  • DOI: https://doi.org/10.1007/978-3-642-31594-7_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-31593-0

  • Online ISBN: 978-3-642-31594-7

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