A Compact Representation for Minimizers of k-Submodular Functions (Extended Abstract)

  • Hiroshi Hirai
  • Taihei OkiEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9849)


k-submodular functions were introduced by Huber and Kolmogorov as a generalization of bisubmodular functions. This paper establishes a compact representation of minimizers of k-submodular functions by posets with inconsistent pairs (PIPs), and completely characterizes the class of PIPs (elementary PIPs) corresponding to minimizers of k-submodular functions. Our representation coincides with Birkhoff’s representation theorem if \(k=1\) and with signed-poset representation by Ando and Fujishige if \(k=2\). We also give algorithms to construct the elementary PIP representing the minimizers of a k-submodular function f for three cases: (i) a minimizing oracle of f is available, (ii) f is network-representable, and (iii) f is the objective function of the relaxation of multiway cut problem. Furthermore, we provide an efficient algorithm to enumerate all maximal minimizers from the PIP representation. Our results are applied to obtain all maximal persistent assignments in labeling problems arising from computer vision.



This work was partially supported by JSPS KAKENHI Grant Numbers 25280004, 26330023, 26280004, and by the JST, ERATO, Kawarabayashi Large Graph Project.


  1. 1.
    Ando, K., Fujishige, S.: \(\sqcup ,\sqcap \)-closed families and signed posets. Technical report, Forschungsinstitut für Diskrete Mathematik, Universität Bonn (1994)Google Scholar
  2. 2.
    Ardila, F., Owen, M., Sullivant, S.: Geodesics in CAT(0) cubical complexes. Adv. Appl. Math. 48(1), 142–163 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Barthélemy, J.P., Constantin, J.: Median graphs, parallelism and posets. Discrete Math. 111(1–3), 49–63 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dahlhaus, E., Johnson, D.S., Papadimitriou, C.H., Seymour, P.D., Yannakakis, M.: The complexity of multiterminal cuts. SIAM J. Comput. 23(4), 864–894 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Feder, T.: Network flow and 2-satisfiability. Algorithmica 11, 291–319 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gridchyn, I., Kolmogorov, V.: Potts model, parametric maxflow and \(k\)-submodular functions. In: IEEE International Conference on Computer Vision (ICCV 2013), pp. 2320–2327 (2013)Google Scholar
  7. 7.
    Hirai, H., Iwamasa, Y.: On \(k\)-submodular relaxation. SIAM J. Discrete Math. (2016, to appear)Google Scholar
  8. 8.
    Huber, A., Kolmogorov, V.: Towards minimizing k-submodular functions. In: Mahjoub, A.R., Markakis, V., Milis, I., Paschos, V.T. (eds.) ISCO 2012. LNCS, vol. 7422, pp. 451–462. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  9. 9.
    Iwata, Y., Wahlström, M., Yoshida, Y.: Half-integrality, LP-branching and FPT algorithms. SIAM J. Comput. (2016, to appear)Google Scholar
  10. 10.
    Kavvadias, D.J., Sideri, M., Stavropoulos, E.C.: Generating all maximal models of a Boolean expression. Inf. Process. Lett. 74, 157–162 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kolmogorov, V., Thapper, J., Živný, S.: The power of linear programming for general-valued CSPs. SIAM J. Comput. 44, 1–36 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Murota, K.: Analysis, Matrices and Matroids for Systems. Springer, Berlin (2010)CrossRefzbMATHGoogle Scholar
  13. 13.
    Nielsen, M., Plotkin, G., Winskel, G.: Petri nets, event structures and domains, part I. Theoret. Comput. Sci. 13, 85–108 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Orlin, J.B.: Max flows in \({\rm O}(nm)\) time, or better. In: Proceedings of the 45th Annual ACM Symposium on Theory of Computing (STOC 2013), pp. 765–774 (2013)Google Scholar
  15. 15.
    Picard, J.C., Queyranne, M.: On the structure of all minimum cuts in a network and applications. In: Rayward-Smith, V.J. (ed.) Combinatorial Optimization II. Mathematical Programming Studies, vol. 13, pp. 8–16. Springer, Berlin (1980)CrossRefGoogle Scholar
  16. 16.
    Sholander, M.: Medians and betweenness. Proc. Am. Math. Soc. 5(5), 801–807 (1954)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Mathematical Informatics, Graduate School of Information Science and TechnologyThe University of TokyoTokyoJapan

Personalised recommendations