Submodular Minimization via Pathwidth

  • Hiroshi Nagamochi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7287)


In this paper, we present a submodular minimization algorithm based on a new relationship between minimizers of a submodular set function and pathwidth defined on submodular set functions. Given a submodular set function f on a finite set V with n ≥ 2 elements and an ordered pair s,t ∈ V, let λ s,t denote the minimum f(X) over all sets X with s ∈ X ⊆ V − {t}. The pathwidth Λ(σ) of a sequence σ of all n elements in V is defined to be the maximum f(V(σ′)) over all nonempty and proper prefixes σ′ of σ, where V(σ′) denotes the set of elements occurred in σ′. The pathwidth Λ s,t of f from s to t is defined to be the minimum pathwidth Λ(σ) over all sequences σ of V which start with element s and end up with t. Given a real k ≥ f({s}), our algorithm checks whether Λ s,t  ≤ k or not and computes λ s,t (when Λ s,t  ≤ k) in O(n Δ(k) + 1) oracle-time, where Δ(k) is the number of distinct values of f(X) with f(X) ≤ k overall sets X with s ∈ X ⊆ V − {t}.


Search Tree Distinct Element Polynomial Time Algorithm Submodular Function Proper Extension 
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  1. 1.
    Arnborg, S., Proskurowski, A.: Linear time algorithms for NP-hard problems restricted to partial k-trees. Discrete Applied Mathematics 23(1), 11–24 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Barát, J.: Directed path-width and monotonicity in digraph searching. Graphs and Combinatorics 22(2), 161–172 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Fleischer, L.K.: Recent progress in submodular function minimization. Optima, 1–11 (2000)Google Scholar
  4. 4.
    Fujishige, S.: Submodular Functions and Optimization, 2nd edn. North-Holland, Amsterdam (2005)zbMATHGoogle Scholar
  5. 5.
    Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid algorithm and its consequences in combinatorial optimization. Combinatorica 1, 499–513 (1981)CrossRefGoogle Scholar
  6. 6.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, Heidelberg (1988)zbMATHCrossRefGoogle Scholar
  7. 7.
    Iwata, S.: Submodular function minimization. Math. Program. 112, 45–64 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Iwata, S., Fleischer, L., Fujishige, S.: A combinatorial, strongly polynomial-time algorithm for minimizing submodular functions. J. ACM 48, 761–777 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Johnson, T., Robertson, N., Seymour, P.D., Thomas, R.: Directed tree-width. Journal of Combinatorial Theory Series B 82(1), 138–154 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Kinnersley, N.G.: The vertex separation number of a graph equals its path-width. Information Processing Letters 42, 345–350 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    McCormick, S.T.: Submodular function minimization. In: Aardal, K., Nemhauser, G., Weismantel, R. (eds.) Discrete Optimization. Handbooks in Operations Research and Management Science, vol. 12. Elsevier, Amsterdam (2005)CrossRefGoogle Scholar
  12. 12.
    Nagamochi, H.: Minimum degree orderings. Algorithmica 56, 17–34 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Nagamochi, H., Ibaraki, T.: A note on minimizing submodular functions. Inf. Proc. Lett. 67, 239–244 (1998)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Nagamochi, H., Ibaraki, T.: Algorithmic Aspects of Graph Connectivity. Cambridge University Press, New York (2008)zbMATHCrossRefGoogle Scholar
  15. 15.
    Orlin, J.B.: A faster strongly polynomial time algorithm for submodular function minimization. Math. Program., Ser. A 118, 237–251 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Queyranne, M.: Minimizing symmetric submodular functions. Math. Program. 82, 3–12 (1998)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Robertson, N., Seymour, P.: Graph minors. I. Excluding a forest. Journal of Combinatorial Theory, Series B 35(1), 39–61 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Robertson, N., Seymour, P.: Graph minors III: Planar tree-width. J. Combin. Theory Ser. B 36(1), 49–64 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Robertson, N., Seymour, P.: Graph Minors. XX. Wagner’s conjecture. J. Combin. Theory Ser. B 92(2), 325–335 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Schrijver, A.: A combinatorial algorithm minimizing submodular functions in strongly polynomial time. J. Combin. Theory Ser. B 80, 346–355 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin (2003)zbMATHGoogle Scholar
  22. 22.
    Tamaki, H.: A Polynomial Time Algorithm for Bounded Directed Pathwidth. In: Kolman, P., Kratochvíl, J. (eds.) WG 2011. LNCS, vol. 6986, pp. 331–342. Springer, Heidelberg (2011)CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Hiroshi Nagamochi
    • 1
  1. 1.Graduate School of InformaticsKyoto UniversityJapan

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