Advertisement

Journal of Combinatorial Optimization

, Volume 36, Issue 3, pp 709–741 | Cite as

A compact representation for minimizers of k-submodular functions

  • Hiroshi Hirai
  • Taihei OkiEmail author
Article
  • 159 Downloads

Abstract

A k-submodular function is a generalization of submodular and bisubmodular functions. This paper establishes a compact representation for minimizers of a k-submodular function by a poset with inconsistent pairs (PIP). This is a generalization of Ando–Fujishige’s signed poset representation for minimizers of a bisubmodular function. We completely characterize the class of PIPs (elementary PIPs) arising from k-submodular functions. We give algorithms to construct the elementary PIP of 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 arises from a Potts energy function. Furthermore, we provide an efficient enumeration algorithm for all maximal minimizers of a Potts k-submodular function. Our results are applicable to obtain all maximal persistent labelings in actual computer vision problems. We present experimental results for real vision instances.

Keywords

k-submodular function Birkhoff representation theorem Poset with inconsistent pairs (PIP) Potts energy function 

Notes

Acknowledgements

We thank Kazuo Murota, Satoru Fujishige, and the referees for helpful comments. This work was partially supported by JSPS KAKENHI Grant Numbers 25280004, 26330023, 26280004, and by JST, ERATO, Kawarabayashi Large Graph Project.

References

  1. Ando K, Fujishige S (1994) \(\sqcup ,\sqcap \)-closed families and signed posets. Technical report, Forschungsinstitut für Diskrete Mathematik, Universität BonnGoogle Scholar
  2. Ardila F, Owen M, Sullivant S (2012) Geodesics in CAT(0) cubical complexes. Adv Appl Math 48:142–163MathSciNetCrossRefzbMATHGoogle Scholar
  3. Babenko MA, Karzanov AV (2012) On weighted multicommodity flows in directed networks. arXiv:1212.0224v1
  4. Barthélemy J-P, Constantin J (1993) Median graphs, parallelism and posets. Discret Math 111(1–3):49–63MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bouchet A (1997) Multimatroids I. Coverings by independent sets. SIAM J Discret Math 10:626–646MathSciNetCrossRefzbMATHGoogle Scholar
  6. Chepoi V (2012) Nice labeling problem for event structures: a counterexample. SIAM J Comput 41:715–727MathSciNetCrossRefzbMATHGoogle Scholar
  7. Feder T (1994) Network flow and 2-satisfiability. Algorithmica 11:291–319MathSciNetCrossRefzbMATHGoogle Scholar
  8. Fujishige S (1995) Submodular functions and combinatorial optimization (in Japanese). In: Proceedings of the 7th research association of mathematical programming symposium (RAMP 1995), pp 13–28Google Scholar
  9. Gridchyn I, Kolmogorov V (2013) Potts model, parametric maxflow and \(k\)-submodular functions. In: Proceedings of the IEEE international conference on computer vision (ICCV 2013), pp 2320–2327Google Scholar
  10. Hirai H (2010) A note on multiflow locking theorem. J Op Res Soc Jpn 53(2):149–156MathSciNetzbMATHGoogle Scholar
  11. Hirai H (2015) L-extendable functions and a proximity scaling algorithm for minimum cost multiflow problem. Discret Optim 18:1–37MathSciNetCrossRefzbMATHGoogle Scholar
  12. Hirai H, Iwamasa Y (2016) On \(k\)-submodular relaxation. SIAM J Discret Math 30:1726–1736MathSciNetCrossRefzbMATHGoogle Scholar
  13. Hirai H, Oki T (2016) A compact representation for minimizers of \(k\)-submodular functions (extended abstract). In: Proceedings of the 4th international symposium on combinatorial optimization (ISCO 2016), volume 9849 of lecture notes in computer science, Springer, Cham, pp 381–392Google Scholar
  14. Huber A, Kolmogorov V (2012) Towards minimizing \(k\)-submodular functions. In: Proceedings of the 2nd international symposium on combinatorial optimization (ISCO 2012), Volume 7422 of lecture notes in computer science, Springer, Heidelberg, pp 451–462Google Scholar
  15. Ibaraki T, Karzanov AV, Nagamochi H (1998) A fast algorithm for finding a maximum free multiflow in an inner eulerian network and some generalizations. Combinatorica 18:61–83MathSciNetCrossRefzbMATHGoogle Scholar
  16. Iwamasa Y (2017) On a general framework for network representability in discrete optimization. J Comb Optim. doi: 10.1007/s10878-017-0136-y
  17. Iwata S, Tanigawa S, Yoshida Y (2016a) Improved approximation algorithms for \(k\)-submodular function maximization. In: Proceedings of the 27th annual ACM-SIAM symposium on discrete algorithms (SODA 2016), pp 404–413Google Scholar
  18. Iwata Y, Wahlström M, Yoshida Y (2016b) Half-integrality, LP-branching and FPT algorithms. SIAM J Comput 45:1377–1411Google Scholar
  19. Kavvadias DJ, Sideri M, Stavropoulos EC (2000) Generating all maximal models of a Boolean expression. Inf Process Lett 74:157–162MathSciNetCrossRefzbMATHGoogle Scholar
  20. Kolmogorov V, Thapper J, Živný S (2015) The power of linear programming for general-valued CSPs. SIAM J Comput 44:1–36MathSciNetCrossRefzbMATHGoogle Scholar
  21. Kovtun I (2003) Partial optimal labeling search for a NP-hard subclass of (max,+) problems. In: Proceedings of the 25th German association for pattern recognition (DAGM 2003), volume 2781 of lecture notes in computer science, Springer, Heidelberg, pp 402–409Google Scholar
  22. Murota K (2000) Matrices and matroids for systems analysis. Springer, BerlinzbMATHGoogle Scholar
  23. Nielsen M, Plotkin G, Winskel G (1981) Petri nets, event structures and domains, part I. Theor Comput Sci 13:85–108CrossRefzbMATHGoogle Scholar
  24. Orlin JB (2013) Max flows in \(\text{O}(nm)\) time, or better. In: Proceedings of the 45th annual ACM Symposium on theory of computing (STOC 2013), pp 765–774Google Scholar
  25. Picard J-C, Queyranne M (1980) On the structure of all minimum cuts in a network and applications. In: Rayward-Smith VJ (ed) Combinatorial optimization II, volume 13 of mathematical programming studies. Springer, Berlin, pp 8–16. doi: 10.1007/BFb0120902
  26. Reiner V (1993) Signed posets. J Comb Theory Ser A 62:324–360MathSciNetCrossRefzbMATHGoogle Scholar
  27. Scharstein D, Szeliski R (2003) High-accuracy stereo depth maps using structured light. In: Proceedings of the 2003 IEEE computer society conference on computer vision and pattern recognition (CVPR 2003), pp 195–202Google Scholar
  28. Scharstein D, Szeliski R, Zabih R (2001) A taxonomy and evaluation of dense two-frame stereo correspondence algorithms. In: Proceedings of the IEEE workshop on stereo and multi-baseline vision (SMBV 2001), pp 131–140Google Scholar
  29. Sholander M (1954) Medians and betweenness. Proc Am Math Soc 5(5):801–807MathSciNetCrossRefzbMATHGoogle Scholar
  30. Squire MB (1995) Enumerating the ideals of a poset. Technical report, North Carolina State UniversityGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

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

Personalised recommendations