Advertisement

Mathematical Programming

, Volume 155, Issue 1–2, pp 1–55 | Cite as

Discrete convexity and polynomial solvability in minimum 0-extension problems

  • Hiroshi HiraiEmail author
Full Length Paper Series A

Abstract

A \(0\)-extension of graph \(\varGamma \) is a metric \(d\) on a set \(V\) containing the vertex set \(V_{\varGamma }\) of \(\varGamma \) such that \(d\) extends the shortest path metric of \(\varGamma \) and for all \(x \in V\) there exists a vertex \(s\) in \(\varGamma \) with \(d(x,s) = 0\). The minimum \(0\)-extension problem 0-Ext \([\varGamma ]\) on \(\varGamma \) is: given a set \(V \supseteq V_{\varGamma }\) and a nonnegative cost function \(c\) defined on the set of all pairs of \(V\), find a \(0\)-extension \(d\) of \(\varGamma \) with \(\sum _{xy}c(xy) d(x,y)\) minimum. The \(0\)-extension problem generalizes a number of basic combinatorial optimization problems, such as minimum \((s,t)\)-cut problem and multiway cut problem. Karzanov proved the polynomial solvability of 0-Ext \([\varGamma ]\) for a certain large class of modular graphs \(\varGamma \), and raised the question: What are the graphs \(\varGamma \) for which 0-Ext \([\varGamma ]\) can be solved in polynomial time? He also proved that 0-Ext \([\varGamma ]\) is NP-hard if \(\varGamma \) is not modular or not orientable (in a certain sense). In this paper, we prove the converse: if \(\varGamma \) is orientable and modular, then 0-Ext \([\varGamma ]\) can be solved in polynomial time. This completes the classification of graphs \(\varGamma \) for which 0-Ext \([\varGamma ]\) is tractable. To prove our main result, we develop a theory of discrete convex functions on orientable modular graphs, analogous to discrete convex analysis by Murota, and utilize a recent result of Thapper and Živný on valued CSP.

Mathematics Subject Classification

90C27 05C12 

Notes

Acknowledgments

We thank the referee for helpful comments, and thank Kazuo Murota for careful reading and numerous helpful comments, Kei Kimura for discussion on Valued-CSP, Satoru Iwata for communicating the paper [43] of Kuivinen, Akiyoshi Shioura for the paper [38] of Kolmogorov, and Satoru Fujishige for the paper [27] of Huber-Kolmogorov. This research is partially supported by the Aihara Project, the FIRST program from JSPS, by Global COE Program “The research and training center for new development in mathematics” from MEXT, and by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

References

  1. 1.
    Bandelt, H.-J.: Networks with condorcet solutions. Eur. J. Oper. Res. 20, 314–326 (1985)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Bandelt, H.-J.: Hereditary modular graphs. Combinatorica 8, 149–157 (1988)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Bandelt, H.-J., Chepoi, V.: Metric graph theory and geometry: a survey. In: Goodman, J.E., Pach, J., Pollack, R. (eds.). Surveys on Discrete and Computational Geometry: Twenty Years Later, pp. 49–86. American Mathematical Society, Providence (2008)Google Scholar
  4. 4.
    Bandelt, H.-J., Chepoi, V., Karzanov, A.V.: A characterization of minimizable metrics in the multifacility location problem. Eur. J. Comb. 21, 715–725 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Bandelt, H.-J., van de Vel, M., Verheul, E.: Modular interval spaces. Mathematische Nachrichten 163, 177–201 (1993)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Birkhoff, G.: Lattice Theory. American Mathematical Society, New York, 1940, 3rd edn. American Mathematical Society, Providence (1967)Google Scholar
  7. 7.
    Bistarelli, S., Montanari, U., Rossi, F.: Semiring-based constraint satisfaction and optimization. J. ACM 44, 201–236 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Bouchet, A.: Multimatroids. I. Coverings by independent sets. SIAM J. Discrete Math. 10, 626–646 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Chandrasekaran, R., Kabadi, S.N.: Pseudomatroids. Discrete Math. 71, 205–217 (1988)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Chalopin, J., Chepoi, V., Hirai, H., Osajda, D.: Weakly modular graphs and nonnegative curvature, Preprint (2014). arXiv:1409.3892
  11. 11.
    Chepoi, V.: Classification of graphs by means of metric triangles. Metody Diskretnogo Analiza 49, 75–93 (1989). (in Russian)MathSciNetGoogle Scholar
  12. 12.
    Chepoi, V.: A multifacility location problem on median spaces. Discrete Appl. Math. 64, 1–29 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Chepoi, V.: Graphs of some \(\text{ CAT }(0)\) complexes. Adv. Appl. Math. 24, 125–179 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Dahlhaus, E., Johnson, D.S., Papadimitriou, C.H., Seymour, P.D., Yannakakis, M.: The complexity of multiterminal cuts. SIAM J. Comput. 23, 864–894 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Dress, A.W.M., Scharlau, R.: Gated sets in metric spaces. Aequationes Mathematicae 34, 112–120 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Fujishige, S.: Submodular Functions and Optimization, 2nd edn. Elsevier, Amsterdam (2005)zbMATHGoogle Scholar
  17. 17.
    Fujishige, S., Murota, K.: Notes on L-/M-convex functions and the separation theorems. Mathematical Programming, Series A 88, 129–146 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Fujishige, S., Tanigawa, S.: A min–max theorem for transversal submodular functions and its implications. SIAM J. Discrete Math (to appear)Google Scholar
  19. 19.
    Fujishige, S., Tanigawa, S., Yoshida, Y.: Generalized skew bisubmodularity: a characterization and a min-max theorem. Discrete Optim. 12, 1–9 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, Berlin (1988)zbMATHCrossRefGoogle Scholar
  21. 21.
    Hirai, H.: Tight spans of distances and the dual fractionality of undirected multiflow problems. J. Comb. Theory Ser. B 99, 843–868 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Hirai, H.: Folder complexes and multiflow combinatorial dualities. SIAM J. Discrete Math. 25, 1119–1143 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Hirai, H.: Half-integrality of node-capacitated multiflows and tree-shaped facility locations on trees. Math. Program. Ser. A 137, 503–530 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Hirai, H.: The maximum multiflow problems with bounded fractionality. Math. Oper. Res. 39, 60–104 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Hirai, H.: Discrete convexity for multiflows and 0-extensions. In: Proceeding of 8th Japanese-Hungarian Symposium on Discrete Mathematics and Its Applications, pp. 209–223 (2013)Google Scholar
  26. 26.
    Hirai, H.: L-convexity on graph structures (in preparation)Google Scholar
  27. 27.
    Huber, A., Kolmogorov, V.: Towards minimizing \(k\)-submodular functions. In: Proceedings of the 2nd International Symposium on Combinatorial Optimization (ISCO’12), LNCS 7422, Springer, Berlin, pp. 451–462 (2012)Google Scholar
  28. 28.
    Huber, A., Krokhin, A.: Oracle tractability of skew bisubmodular functions. SIAM J. Discrete Math. 28, 1828–1837 (2014)Google Scholar
  29. 29.
    Huber, A., Krokhin, A., Powell, R.: Skew bisubmodularity and valued CSPs. SIAM J. Comput. 43, 1064–1084 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Iwata, S., Fleischer, L., Fujishige, S.: A combinatorial strongly polynomial algorithm for minimizing submodular functions. J. ACM 48, 761–777 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Karzanov, A.V.: Polyhedra related to undirected multicommodity flows. Linear Algebra Appl. 114(115), 293–328 (1989)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Karzanov, A.V.: Minimum \(0\)-extensions of graph metrics. Eur. J. Comb. 19, 71–101 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Karzanov, A.V.: Metrics with finite sets of primitive extensions. Ann. Comb. 2, 211–241 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Karzanov, A.V.: One more well-solved case of the multifacility location problem. Discrete Optim. 1, 51–66 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Karzanov, A.V.: Hard cases of the multifacility location problem. Discrete Appl. Math. 143, 368–373 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Kleinberg, J., Tardos, É.: Approximation algorithms for classification problems with pairwise relationships: metric labeling and Markov random fields. J. ACM 49, 616–639 (2002)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Kolen, A.W.J.: Tree Network and Planar Rectilinear Location Theory, CWI Tract 25. Center for Mathematics and Computer Science, Amsterdam (1986)Google Scholar
  38. 38.
    Kolmogorov, V.: Submodularity on a tree: unifying L\(^\natural \)-convex and bisubmodular functions. In: Proceedings of the 36th International Symposium on Mathematical Foundations of Computer Science (MFCS’11), LNCS 6907, Springer, Berlin, pp. 400–411 (2011)Google Scholar
  39. 39.
    Kolmogorov, V.: The power of linear programming for finite-valued CSPs: a constructive characterization. In: Proceedings of the 40th International Colloquium, ICALP 2013, LNCS 7965, pp. 625–636 (2013)Google Scholar
  40. 40.
    Kolmogorov, V., Shioura, A.: New algorithms for convex cost tension problem with application to computer vision. Discrete Optim. 6, 378–393 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  41. 41.
    Kolmogorov, V., Thapper, J., Živný, S.: The power of linear programming for general-valued CSPs, Preprint (2013). arXiv:1311.4219
  42. 42.
    Kolmogorov, V., Živný, S.: The complexity of conservative valued CSPs. J. ACM 60. Article No. 10 (2013)Google Scholar
  43. 43.
    Kuivinen, F.: On the complexity of submodular function minimisation on diamonds. Discrete Optim. 8, 459–477 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    Lovász, L.: Submodular functions and convexity. In: Bachem, A., Grötschel, M., Korte, B. (eds.) Mathematical Programming—The State of the Art, pp. 235–257. Springer, Berlin (1983)CrossRefGoogle Scholar
  45. 45.
    Murota, K.: Discrete convex analysis. Math. Program. 83, 313–371 (1998)zbMATHMathSciNetGoogle Scholar
  46. 46.
    Murota, K.: Algorithms in discrete convex analysis. IEICE Trans. Syst. Inf. E83-D, 344–352 (2000)Google Scholar
  47. 47.
    Murota, K.: Discrete Convex Analysis. SIAM, Philadelphia (2003)zbMATHCrossRefGoogle Scholar
  48. 48.
    Murota, K., Shioura, A.: M-convex function on generalized polymatroid. Math. Oper. Res. 24, 95–105 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  49. 49.
    Murota, K., Tamura, A.: Proximity theorems of discrete convex functions. Math. Program. Ser. A 99, 539–562 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  50. 50.
    Nakamura, M.: A characterization of greedy sets: universal polymatroids (I). Sci. Pap. Coll. Arts Sci. Univ. Tokyo 38, 155–167 (1988)Google Scholar
  51. 51.
    Picard, J.C., Ratliff, D.H.: A cut approach to the rectilinear distance facility location problem. Oper. Res. 26, 422–433 (1978)zbMATHMathSciNetCrossRefGoogle Scholar
  52. 52.
    Qi, L.: Directed submodularity, ditroids and directed submodular flows. Math. Program. 42, 579–599 (1988)zbMATHCrossRefGoogle Scholar
  53. 53.
    Schiex, T., Fargier, H., Verfaillie, G.: Valued constraint satisfaction problems: hard and easy problems. In: Proceedings of the 14th International Joint Conference on Artificial Intelligence (IJCAI’95) (1995)Google Scholar
  54. 54.
    Schrijver, A.: A combinatorial algorithm minimizing submodular functions in strongly polynomial time. J. Comb. Theory Ser. B 80, 346–355 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  55. 55.
    Tansel, B.C., Francis, R.L., Lowe, T.J.: Location on networks I, II. Manag. Sci. 29, 498–511 (1983)zbMATHMathSciNetCrossRefGoogle Scholar
  56. 56.
    Thapper, J., Živný, S.: The power of linear programming for valued CSPs. In: Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS’12), pp. 669–678 (2012)Google Scholar
  57. 57.
    Thapper, J., Živný, S.: The complexity of finite-valued CSPs. In: Proceedings of the 45th ACM Symposium on the Theory of Computing (STOC’13), pp. 695–704 (2013)Google Scholar
  58. 58.
    van de Vel, M.L.J.: Theory of Convex Structures. North-Holland, Amsterdam (1993)zbMATHGoogle Scholar
  59. 59.
    Živný, S.: The Complexity of Valued Constraint Satisfaction Problems. Springer, Heidelberg (2012)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2014

Authors and Affiliations

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

Personalised recommendations