Techniques for Submodular Maximization

Part of the Fields Institute Communications book series (FIC, volume 69)


Maximization of a submodular function is a central problem in the algorithmic theory of combinatorial optimization. On the one hand, it has the feel of a clean and stylized problem, amenable to mathematical analysis, while on the other hand, it comfortably contains several rather different problems which are independently of interest from both theoretical and applied points of view. There have been successful analyses from the point of view of theoretical computer science, specifically approximation algorithms, and from an operations research viewpoint, specifically novel branch-and-bound methods have proven to be effective on broad subclasses of problems. To some extent, both of these points of view have validated what some practitioners have known all along: Local-search methods are very effective for many of these problems.

Key words

Combinatorial optimization Local search Maximum entropy sampling Max cut 

Subject Classifications

90C27 65K05 94A17 62K05 



Partially supported by NSF Grant CMMI–1160915.


  1. 1.
    Anstreicher, K.M., Lee, J.: A masked spectral bound for maximum-entropy sampling. In: MODA 7 – Advances in Model-Oriented Design and Analysis. Contributions to Statistics, pp. 1–12. Springer, Berlin (2004)Google Scholar
  2. 2.
    Anstreicher, K.M., Fampa, M., Lee, J., Williams, J.D.: Continuous relaxations for constrained maximum-entropy sampling. In: Integer Programming and Combinatorial Optimization, Vancouver. Lecture Notes in Computer Science, vol. 1084, pp. 234–248. Springer, Berlin (1996)Google Scholar
  3. 3.
    Anstreicher, K.M., Fampa, M., Lee, J., Williams, J.D.: Using continuous nonlinear relaxations to solve constrained maximum-entropy sampling problems. Math. Program. A 85, 221–240 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Anstreicher, K.M., Fampa, M., Lee, J., Williams, J.D.: Maximum-entropy remote sampling. Discret. Appl. Math. 108, 211–226 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Barahona, F., Grötschel, M., Jünger, M., Reinelt, G.: An application of combinatorial optimization to statistical physics and circuit layout design. Oper. Res. 36(3), 493–513 (1988)zbMATHCrossRefGoogle Scholar
  6. 6.
    Burer, S., Lee, J.: Solving maximum-entropy sampling problems using factored masks. Math. Program. 109(2–3), 263–281 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Calinescu, G., Chekuri, C., Pál, M., Vondrák, J.: Maximizing a submodular set function subject to a matroid constraint. In: Proceedings of 12th IPCO, Ithaca, pp. 182–196 (2007)Google Scholar
  8. 8.
    Caselton, W.F., Zidek, J.V.: Optimal monitoring network designs. Stat. Probab. Lett. 2, 223–227 (1984)zbMATHCrossRefGoogle Scholar
  9. 9.
    Caselton, W.F., Kan, L., Zidek, J.V.: Quality data networks that minimize entropy. In: Walden, A.T., Guttorp, P. (eds.) Statistics in the Environmental and Earth Sciences, pp. 10–38. Arnold, London (1992)Google Scholar
  10. 10.
    D’Ambrosio, C., Lee, J., Wächter, A.: An algorithmic framework for MINLP with separable non-convexity. In: Leyffer, S., Lee, J. (eds.) Mixed Integer Nonlinear Programming. The IMA Volumes in Mathematics and Its Applications, vol. 154, pp.315–347. Springer, New York (2012)CrossRefGoogle Scholar
  11. 11.
    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
  12. 12.
    Frank, A.: Applications of submodular functions. In: Walker, K. (ed.) Surveys in Combinatorics. London Mathematical Society, Lecture Note Series, vol. 187, pp.850–136. Cambridge University Press, Cambridge (1993)Google Scholar
  13. 13.
    Fujishige, S.: Submodular functions and optimization. Annals of Discrete Mathematics, vol. 58, 2nd edn. Elsevier, Amsterdam (2005)Google Scholar
  14. 14.
    Fujishige, S., Isotani, S.: A submodular function minimization algorithm based on the minimum-norm base. Pac. J. Optim. 7, 3–17 (2011)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42(6), 1115–1145 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequence in combinatorial optimization. Combinatorica 1(2), 169–197 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, Berlin (1993)zbMATHCrossRefGoogle Scholar
  18. 18.
    Guttorp, P., Le, N.D., Sampson, P.D., Zidek, J.V.: Using Entropy in the redesign of an environmental monitoring network. In: Patil, G.P., Rao, C.R. (eds.) Multivariate Environmental Statistics, pp. 175–202. North Holland, New York (1993)Google Scholar
  19. 19.
    Hoffman, A.J., Lee, J., Williams, J.D.: New upper bounds for maximum-entropy sampling. In: Atkinson, A.C., Hackl, P., Müller, W.G. (eds.) MODA 6 – Advances in Model-Oriented Design and Analysis. Contributions to Statistics, pp. 143–153. Springer, Berlin (2001)CrossRefGoogle Scholar
  20. 20.
    Iwata, S., Fleischer, L., Fujishige, S.: A combinatorial, strongly polynomial-time algorithm for minimizing submodular functions. In: Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, Portland, pp. 97–106 (electronic). ACM, New York (2000)Google Scholar
  21. 21.
    Kelmans, A.K., Kimelfeld, B.N.: Multiplicative submodularity of a matrix?s principal minor as a function of the set of its rows and some combinatorial applications. Discret. Math. 44(1), 113–116 (1980)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Ko, C.-W., Lee, J., Queyranne, M.: An exact algorithm for maximum entropy sampling. Oper. Res. 43(4), 684–691 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Krause, A.: SFO: a toolbox for submodular function optimization. J. Mach. Learn. Res. 11, 1141–1144 (2010)zbMATHGoogle Scholar
  24. 24.
    Krause, A., Guestrin, C.: Beyond convexity: submodularity in machine learning (2013).
  25. 25.
    Laurent, M.: Semidefinite relaxations for max-cut. In: The Sharpest Cut. MPS/SIAM Series on Optimization, pp. 257–290. SIAM, Philadelphia (2004)Google Scholar
  26. 26.
    Le, N.D., Zidek, J.V.: Statistical analysis of environmental space-time processes. Springer Series in Statistics. Springer, New York (2006)zbMATHGoogle Scholar
  27. 27.
    Lee, J.: Discussion on: ‘A state-space-model approach to optimal spatial sampling design based on entropy’. Environ. Ecol. Stat. 5, 45–46 (1998)CrossRefGoogle Scholar
  28. 28.
    Lee, J.: Constrained maximum-entropy sampling. Oper. Res. 46, 655–664 (1998)zbMATHCrossRefGoogle Scholar
  29. 29.
    Lee, J.: Semidefinite programming in experimental design. In: Wolkowicz, H., Saigal, R., Vandenberghe, L. (eds.) Handbook of Semidefinite Programming. International Series in Operations Research and Management Science, vol. 27. Kluwer, Boston (2000)Google Scholar
  30. 30.
    Lee, J.: Maximum entropy sampling. In: El-Shaarawi, A.H., Piegorsch, W.W. (eds.) Encyclopedia of Environmetrics, 2nd edn., vol. 3, pp. 1570–1574. Wiley, Chichester (2012)Google Scholar
  31. 31.
    Lee, J.: A First Course in Combinatorial Optimization. Cambridge University Press, Cambridge/New York (2004)zbMATHCrossRefGoogle Scholar
  32. 32.
    Lee, J., Williams, J.: A linear integer programming bound for maximum-entropy sampling. Math. Program. B 94, 247–256 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Lee, J., Sviridenko, M., Vondrák, J.: Submodular maximization over multiple matroids via generalized exchange properties. In: Dinur, I., Jansen, K., Naor, S., Rolim, J.D.P. (eds.) Proceedings of APPROX 2009, 12th International Workshop, Berkeley, 21–23 Aug 2009. Lecture Notes in Computer Science, vol. 5687, pp. 244–257. Springer (2009)Google Scholar
  34. 34.
    Lee, J., Mirrokni, V.S., Nagarajan, V., Sviridenko, M.: Non-monotone submodular maximization under matroid and knapsack constraints. In: Proceedings of the 41st Annual ACM Symposium on Theory of Computing STOC 2009, Bethesda, pp. 323–332 (2009)Google Scholar
  35. 35.
    Lee, J., Sviridenko, M., Vondrák, J.: Approximate maximization of a submodular function over multiple matroids via generalized exchange properties. Math. Oper. Res. 35(4), 795–806 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Lee, J., Mirrokni, V.S., Nagarajan, V., Sviridenko, M.: Maximizing non-monotone submodular functions under matroid and knapsack constraints. SIAM J. Discret. Math. 23(4), 2053–2078 (2010)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Lehmann, B., Lehmann, D.J., Nisan, N.: Combinatorial auctions with decreasing marginal utilities (journal version). Games Econ. Behav. 55, 270–296 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    NADP/NTN website (2013).
  39. 39.
    Narayanan, H.: Submodular functions and electrical networks. Annals of Discrete Mathematics, vol. 54. North-Holland, Amsterdam (1997)Google Scholar
  40. 40.
    Nemhauser, G.L., Wolsey, L.A., Fisher, M.: An analysis of approximations for maximizing submodular set functions. I. Math. Program. 14(3), 265–294 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Nemhauser, G.L., Wolsey, L.A., Laurence A.: Integer and Combinatorial Optimization. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York (1988)zbMATHGoogle Scholar
  42. 42.
    Oppenheim, A.: Inequalities connected with definite Hermitian forms. J. Lond. Math. Soc. 5, 114–119 (1930)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Poljak, S., Rendl, F.: Solving the max-cut problem using eigenvalues. Special volume on partitioning and decomposition in combinatorial optimization. Discret. Appl. Math. 62(1–3), 249–278 (1995)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Recski, A.: Matroid theory and its applications in electric network theory and in statics. Algorithms and Combinatorics, vol. 6. Springer, Berlin (1989)Google Scholar
  45. 45.
    Rendl, F., Rinaldi, G., Wiegele, A.: Solving max-cut to optimality by intersecting semidefinite and polyhedral relaxations. Math. Program. A 121 (2), 307–335 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Rendl, F., Rinaldi, G., Wiegele, A.: Biq mac solver – binary quadratic and max cut solver (2013).
  47. 47.
    Schrijver, A.: A combinatorial algorithm minimizing submodular functions in strongly polynomial time. J. Comb. Theory B 80(2), 346–355 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Semple, C.: Submodular functions and biodiversity conservation (2010).
  49. 49.
    Shewry, M.C., Wynn, H.P.: Maximum entropy sampling. J. Appl. Stat. 14, 165–170 (1987)CrossRefGoogle Scholar
  50. 50.
    Sviridenko, M.: A note on maximizing a submodular set function subject to knapsack constraint. Oper. Res. Lett. 32, 41–43 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Wu, S., Zidek, J.V.: An entropy based review of selected NADP/NTN network sites. Atmos. Environ. 26A, 2089–2103 (1992)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  1. 1.IOE DepartmentUniversity of MichiganAnn ArborUSA

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