Skip to main content
SpringerLink
Log in

Submodularity on a Tree: Unifying \(L^\natural\)-Convex and Bisubmodular Functions

  • Conference paper
Mathematical Foundations of Computer Science 2011 (MFCS 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6907))

Abstract

We introduce a new class of functions that can be minimized in polynomial time in the value oracle model. These are functions f satisfying \(f(\mbox{\boldmath $x$})+f(\mbox{\boldmath $y$})\ge f(\mbox{\boldmath $x$} \sqcap \mbox{\boldmath $y$})+f(\mbox{\boldmath $x$} \sqcup \mbox{\boldmath $y$})\) where the domain of each variable x i corresponds to nodes of a rooted binary tree, and operations ⊓ , ⊔ are defined with respect to this tree. Special cases include previously studied \(L^\natural\)-convex and bisubmodular functions, which can be obtained with particular choices of trees. We present a polynomial-time algorithm for minimizing functions in the new class. It combines Murota’s steepest descent algorithm for \(L^\natural\)-convex functions with bisubmodular minimization algorithms.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Barto, L., Kozik, M., Niven, T.: The CSP dichotomy holds for digraphs with no sources and no sinks (a positive answer to a conjecture of Bang-Jensen and Hell). SIAM Journal on Computing 38(5), 1782–1802 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Birkhoff, G.: Rings of sets. Duke Mathematical Journal 3(3), 443–454 (1937)

    Article  MathSciNet  Google Scholar 

  3. Bouchet, A.: Greedy algorithm and symmetric matroids. Math. Programming 38, 147–159 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bouchet, A., Cunningham, W.H.: Delta-matroids, jump systems and bisubmodular polyhedra. SIAM J. Discrete Math. 8, 17–32 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bulatov, A.A.: Tractable Conservative Constraint Satisfaction Problems. In: Proceedings of the 18th IEEE Symposium on Logic in Computer Science (LICS 2003), pp. 321–330 (2003)

    Google Scholar 

  6. Bulatov, A.A.: A dichotomy theorem for constraint satisfaction problems on a 3-element set. Journal of the ACM 53(1), 66–120 (2006)

    Article  MathSciNet  Google Scholar 

  7. Chandrasekaran, R., Kabadi, S.N.: Pseudomatroids. Discrete Math. 71, 205–217 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cohen, D., Cooper, M., Jeavons, P.G.: A Complete Characterization of Complexity for Boolean Constraint Optimization Problems. In: Wallace, M. (ed.) CP 2004. LNCS, vol. 3258, pp. 212–226. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  9. Cohen, D.A., Cooper, M., Jeavons, P.G., Krokhin, A.A.: Soft Constraints: Complexity and Multimorphisms. In: Rossi, F. (ed.) CP 2003. LNCS, vol. 2833, pp. 244–258. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  10. Cohen, D.A., Cooper, M.C., Jeavons, P.G.: Generalising submodularity and horn clauses: Tractable optimization problems defined by tournament pair multimorphisms. Theoretical Computer Science 401, 36–51 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cohen, D.A., Cooper, M.C., Jeavons, P.G., Krokhin, A.A.: The complexity of soft constraint satisfaction. Artificial Intelligence 170, 983–1016 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Deineko, V., Jonsson, P., Klasson, M., Krokhin, A.: The approximability of Max CSP with fixed-value constraints. Journal of the ACM 55(4) (2008)

    Google Scholar 

  13. Favati, P., Tardella, F.: Convexity in nonlinear integer programming. Ricerca Operativa 53, 3–44 (1990)

    Google Scholar 

  14. Feder, T., Vardi, M.: The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory. SIAM Journal on Computing 28(1), 57–104 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fujishige, S.: Submodular Functions and Optimization. North-Holland, Amsterdam (1991)

    MATH  Google Scholar 

  16. Fujishige, S., Murota, K.: Notes on L-/M-convex functions and the separation theorems. Math. Program. 88, 129–146 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fujishige, S., Iwata, S.: Bisubmodular function minimization. SIAM J. Discrete Math. 19(4), 1065–1073 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, Heidelberg (1988)

    MATH  Google Scholar 

  19. Iwata, S., Fleischer, L., Fujishige, S.: A combinatorial strongly polynomial algorithm for minimizing submodular functions. J. ACM 48, 761–777 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jonsson, P., Kuivinen, F., Thapper, J.: Min CSP on Four Elements: Moving Beyond Submodularity. Tech. rep. arXiv:1102.2880 (February 2011)

    Google Scholar 

  21. Kabadi, S.N., Chandrasekaran, R.: On totally dual integral systems. Discrete Appl. Math. 26, 87–104 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kolmogorov, V., Shioura, A.: New algorithms for convex cost tension problem with application to computer vision. Discrete Optimization 6(4), 378–393 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kolmogorov, V., Živný, S.: The complexity of conservative finite-valued CSPs. Tech. rep. arXiv:1008.1555v1 (August 2010)

    Google Scholar 

  24. Kolmogorov, V., Živný, S.: Generalising tractable VCSPs defined by symmetric tournament pair multimorphisms. Tech. rep. arXiv:1008.3104v1 (August 2010)

    Google Scholar 

  25. Kolmogorov, V.: A dichotomy theorem for conservative general-valued CSPs. Tech. rep. arXiv:1008.4035v1 (August 2010)

    Google Scholar 

  26. Kolmogorov, V.: Submodularity on a tree: Unifying \(L^\natural\)-convex and bisubmodular functions. Tech. rep. arXiv:1007.1229v3, April 2011 (first version: July 2010)

    Google Scholar 

  27. Krokhin, A., Larose, B.: Maximizing supermodular functions on product lattices, with application to maximum constraint satisfaction. SIAM Journal on Discrete Mathematics 22(1), 312–328 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kuivinen, F.: On the Complexity of Submodular Function Minimisation on Diamonds. Tech. rep. arXiv:0904.3183v1 (April 2009)

    Google Scholar 

  29. McCormick, S.T., Fujishige, S.: Strongly polynomial and fully combinatorial algorithms for bisubmodular function minimization. Math. Program., Ser. A 122, 87–120 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Murota, K.: Discrete convex analysis. Math. Program. 83, 313–371 (1998)

    MathSciNet  MATH  Google Scholar 

  31. Murota, K.: Algorithms in discrete convex analysis. IEICE Transactions on Systems and Information E83-D, 344–352 (2000)

    Google Scholar 

  32. Murota, K.: Discrete Convex Analysis. SIAM Monographs on Discrete Mathematics and Applications 10 (2003)

    Google Scholar 

  33. Murota, K.: On steepest descent algorithms for discrete convex functions. SIAM J. Optimization 14(3), 699–707 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  34. Nakamura, M.: A characterization of greedy sets: universal polymatroids (I). Scientific Papers of the College of Arts and Sciences 38(2), 155–167 (1998); The University of Tokyo

    Google Scholar 

  35. Qi, L.: Directed submodularity, ditroids and directed submodular flows. Mathematical Programming 42, 579–599 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  36. Schaefer, T.: The Complexity of Satisfiability Problems. In: Proceedings of the 10th Annual ACM Symposium on Theory of Computing (STOC 1978), pp. 216–226 (1978)

    Google Scholar 

  37. Schrijver, A.: A combinatorial algorithm minimizing submodular functions in strongly polynomial time. J. Combin. Theory Ser. B 80, 346–355 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  38. Takhanov, R.: A Dichotomy Theorem for the General Minimum Cost Homomorphism Problem. In: Proceedings of the 27th International Symposium on Theoretical Aspects of Computer Science (STACS 2010), pp. 657–668 (2010)

    Google Scholar 

  39. Topkis, D.M.: Minimizing a submodular function on a lattice. Operations Research 26(2), 305–321 (1978)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University College London, UK

    Vladimir Kolmogorov

Authors
  1. Vladimir Kolmogorov
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute of Informatics, University of Warsaw, ul. Banacha 2, 02-097, Warsaw, Poland

    Filip Murlak  & Piotr Sankowski  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag GmbH Berlin Heidelberg

About this paper

Cite this paper

Kolmogorov, V. (2011). Submodularity on a Tree: Unifying \(L^\natural\)-Convex and Bisubmodular Functions. In: Murlak, F., Sankowski, P. (eds) Mathematical Foundations of Computer Science 2011. MFCS 2011. Lecture Notes in Computer Science, vol 6907. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22993-0_37

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-22993-0_37

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-22992-3

  • Online ISBN: 978-3-642-22993-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation