Advertisement

Towards Minimizing k-Submodular Functions

  • Anna Huber
  • Vladimir Kolmogorov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7422)

Abstract

In this paper we investigate k-submodular functions. This natural family of discrete functions includes submodular and bisubmodular functions as the special cases k = 1 and k = 2 respectively.

In particular we generalize the known Min-Max-Theorem for submodular and bisubmodular functions. This theorem asserts that the minimum of the (bi)submodular function can be found by solving a maximization problem over a (bi)submodular polyhedron. We define a k-submodular polyhedron, prove a Min-Max-Theorem for k-submodular functions, and give a greedy algorithm to construct the vertices of the polyhedron.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bouchet, A.: Greedy algorithm and symmetric matroids. Math. Progr. 38, 147–159 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bouchet, A.: Multimatroids I. coverings by independent sets. SIAM J. Discrete Math. 10(4), 626–646 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bouchet, A.: Multimatroids II. orthogonality, minors and connectivity. Electr. J. Comb. 5 (1998)Google Scholar
  4. 4.
    Bouchet, A.: Multimatroids III. tightness and fundamental graphs. Eur. J. Comb. 22(5), 657–677 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bouchet, A., Cunningham, W.H.: Delta-matroids, jump systems and bisubmodular polyhedra. SIAM J. Discrete Math. 8, 17–32 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Chandrasekaran, R., Kabadi, S.N.: Pseudomatroids. Disc. Math. 71, 205–217 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Cohen, D., Cooper, M., Jeavons, P.: Generalising submodularity and Horn clauses: Tractable optimization problems defined by tournament pair multimorphisms. Theoretical Computer Science 401(1), 36–51 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Cohen, D., Cooper, M., Jeavons, P., Krokhin, A.: The complexity of soft constraint satisfaction. Artificial Intelligence 170(11), 983–1016 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Cunningham, W.H., Green-Krótki, J.: b-matching degree-sequence polyhedra. Combinatorica 11(3), 219–230 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Deineko, V., Jonsson, P., Klasson, M., Krokhin, A.: The approximability of max CSP with fixed-value constraints. J. ACM 55(4) (2008)Google Scholar
  11. 11.
    Dunstan, F.D.J., Welsh, D.J.A.: A greedy algorithm for solving a certain class of linear programmes. Math. Progr. 5, 338–353 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Edmonds, J.: Submodular functions, matroids, and certain polyhedra. In: Guy, R., Hanani, H., Sauer, N., Schönheim, J. (eds.) Combinatorial Structures and Their Applications, pp. 69–87. Gordon and Breach (1970)Google Scholar
  13. 13.
    Frank, A.: Applications of submodular functions. In: Walker, K. (ed.) Surveys in Combinatorics, pp. 85–136. Cambridge University Press (1993)Google Scholar
  14. 14.
    Fujishige, S.: A min-max theorem for bisubmodular polyhedra. SIAM J. Discrete Math. 10(2), 294–308 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Fujishige, S.: Submodular Functions and Optimization. Elsevier (2005)Google Scholar
  16. 16.
    Fujishige, S., Iwata, S.: Bisubmodular function minimization. SIAM J. Discrete Math. 19(4), 1065–1073 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1, 169–197 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Iwata, S.: Submodular function minimization. Math. Progr. 112(1), 45–64 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Iwata, S., Fleischer, L., Fujishige, S.: A combinatorial strongly polynomial algorithm for minimizing submodular functions. J. ACM 48(4), 761–777 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Iwata, S., Orlin, J.: A simple combinatorial algorithm for submodular function minimization. In: SODA, pp. 1230–1237 (2009)Google Scholar
  21. 21.
    Jonsson, P., Kuivinen, F., Thapper, J.: Min CSP on Four Elements: Moving beyond Submodularity. In: Lee, J. (ed.) CP 2011. LNCS, vol. 6876, pp. 438–453. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  22. 22.
    Kabadi, S.N., Chandrasekaran, R.: On totally dual integral systems. Discrete Appl. Math. 26, 87–104 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Kolmogorov, V.: Submodularity on a Tree: Unifying \(L^\natural\)-Convex and Bisubmodular Functions. In: Murlak, F., Sankowski, P. (eds.) MFCS 2011. LNCS, vol. 6907, pp. 400–411. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  24. 24.
    Kolmogorov, V., Zivny, S.: The complexity of conservative valued CSPs. In: SODA (2012)Google Scholar
  25. 25.
    Krokhin, A., Larose, B.: Maximizing supermodular functions on product lattices, with application to maximum constraint satisfaction. SIAM J. Discrete Math. 22(1), 312–328 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Kuivinen, F.: On the complexity of submodular function minimisation on diamonds. Discrete Optimization 8(3), 459–477 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Lovász, L.: Submodular functions and convexity. In: Bachem, A., Grötschel, M., Korte, B. (eds.) Math. Progr.: The State of the Art, pp. 235–257 (1983)Google Scholar
  28. 28.
    McCormick, S.T., Fujishige, S.: Strongly polynomial and fully combinatorial algorithms for bisubmodular function minimization. Math. Progr. 122, 87–120 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    McCormick, S.: Submodular function minimization. In: Aardal, K., Nemhauser, G., Weismantel, R. (eds.) Handbook on Discr. Opt., pp. 321–391. Elsevier (2006)Google Scholar
  30. 30.
    Nakamura, M.: A characterization of greedy sets: universal polymatroids (I). Scientific Papers of the College of Arts and Sciences 38, 155–167 (1998)Google Scholar
  31. 31.
    Orlin, J.: A faster strongly polynomial time algorithm for submodular function minimization. Math. Progr. 118, 237–251 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Qi, L.: Directed submodularity, ditroids and directed submodular flows. Math. Progr. 42, 579–599 (1988)zbMATHCrossRefGoogle Scholar
  33. 33.
    Raghavendra, P.: Approximating NP-hard Problems: Efficient Algorithms and their Limits. PhD Thesis (2009)Google Scholar
  34. 34.
    Schrijver, A.: A combinatorial algorithm minimizing submodular functions in polynomial time. Journal of Combinatorial Theory, Ser. B 80, 346–355 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency (2004)Google Scholar
  36. 36.
    Takhanov, R.: A dichotomy theorem for the general minimum cost homomorphism problem. In: STACS, pp. 657–668 (2010)Google Scholar
  37. 37.
    Thapper, J., Živný, S.: The Power of Linear Programming for Valued CSPs. ArXiv abs/1204.1079 (2012)Google Scholar
  38. 38.
    Topkis, D.M.: Minimizing a submodular function on a lattice. Operations Research 26(2), 305–321 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Topkis, D.M.: Supermodularity and complementarity. Princeton Univ. Press (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Anna Huber
    • 1
  • Vladimir Kolmogorov
    • 2
  1. 1.School of Engineering and Computing SciencesDurham UniversityDurhamUK
  2. 2.Institute of Science and Technology AustriaKlosterneuburgAustria

Personalised recommendations