Skip to main content
SpringerLink
Log in

Minimization of Locally Defined Submodular Functions by Optimal Soft Arc Consistency

  • Published:
Constraints Aims and scope Submit manuscript

Abstract

Submodular function minimization is a polynomially solvable combinatorial problem. Unfortunately the best known general-purpose algorithms have high-order polynomial time complexity. In many applications the objective function is locally defined in that it is the sum of cost functions (also known as soft or valued constraints) whose arities are bounded by a constant. We prove that every valued constraint satisfaction problem with submodular cost functions has an equivalent instance on the same constraint scopes in which the actual minimum value of the objective function is rendered explicit. Such an equivalent instance is the result of establishing optimal soft arc consistency and can hence be found by solving a linear program. From a practical point of view, this provides us with an alternative algorithm for minimizing locally defined submodular functions. From a theoretical point of view, this brings to light a previously unknown connection between submodularity and soft arc consistency.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bessière, C., & Régin, J.-C. (2001). Refining the basic constraint propagation algorithm. In Proc IJCAI’01 (pp. 309–315). Seattle, WA.

  2. Billionet, A., & Minoux, M. (1985). Maximizing a supermodular pseudo-boolean function: A polynomial algorithm for cubic functions. Discrete Applied Mathematics, 12, 1–11.

    Article  MathSciNet  Google Scholar 

  3. Boros, E., & Hammer, P. L. (2002). Pseudo-boolean optimization. Discrete Applied Mathematics, 123, 155–225.

    Article  MATH  MathSciNet  Google Scholar 

  4. Bulatov, A. A. (2002). A dichotomy theorem for constraints on a three-element set. In Proc. 43rd IEEE symposium on foundations of computer science (FOCS’02) (pp. 649–658).

  5. Bulatov, A. A. (2003). Tractable conservative constraint satisfaction problems. In Proc. 18th IEEE symposium on logic in computer science (LICS’03) (pp. 321–330).

  6. Bulatov, A. A. (2006). Combinatorial problems raised from 2-semilattices. Journal of Algebra, 321–339.

  7. Bulatov, A. A., Krokhin, A. A., & Jeavons, P. G. (2005). Classifying the complexity of constraints using finite algebras. SIAM Journal on Computing, 34, 720–742.

    Article  MATH  MathSciNet  Google Scholar 

  8. Cohen, D., Cooper, M. C., & Jeavons, P. (2004). A complete characterisation of complexity for Boolean constraint optimization problems., In Proc. 10th Int. Conf. on Priciples and Practice of Constraint Programming (CP’04), LNCS 3258 (pp. 212–226).

  9. Cohen, D., Cooper, M. C., Jeavons, P., & Krokhin, A. (2004). A maximal tractable class of soft constraints. Journal of Artificial Intelligence Research, 22, 1–22.

    Article  MATH  MathSciNet  Google Scholar 

  10. Cohen, D., Cooper, M. C., Jeavons, P., & Krokhin, A. (2005). Supermodular functions and the complexity of Max-CSP. Discrete Applied Mathematics, 149, 53–72.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  12. Cohen, D., Cooper, M. C., & Jeavons, P. (2006). An algebraic characterisation of complexity for valued constraints. In Proc. CP’06, LNCS 4204 (pp. 107–121).

  13. Cohen, D., Cooper, M.C., & Jeavons, P. (2008). Generalising submodularity and horn clauses: Tractable optimization problems defined by tournament pair multimorphisms. Theoretical Computer Science (in press)

  14. Cooper, M. C. (2003). Reduction operations in fuzzy and valued constraint satisfaction. Fuzzy Sets and Systems, 134, 311–342.

    Article  MATH  MathSciNet  Google Scholar 

  15. Cooper, M. C. (2005). High-order consistency in valued constraint satisfaction. Constraints, 10, 283–305.

    Article  MATH  MathSciNet  Google Scholar 

  16. Cooper, M. C., de Givry, S., & Schiex, T. (2007). Optimal soft arc consistency. In Proc. IJCAI’07 (pp. 68–73). Hyderabad.

  17. Cooper, M. C., & Schiex, T. (2004). Arc consistency for soft constraints. Artificial Intelligence, 154(1–2), 199–227.

    Article  MATH  MathSciNet  Google Scholar 

  18. Creignou, N. (1995). A dichotomy theorem for maximum generalised satisfiability problems. Journal of Computer and Systems Sciences, 51(3), 511–522.

    Article  MathSciNet  Google Scholar 

  19. Creignou, N., Khanna, S., & Sudan, M. (2001). Complexity classification of Boolean constraint satisfaction problems. In SIAM monographs on discrete mathematics and applications 7.

  20. Cunningham, W. H. (1985). Minimum cuts, modular functions, and matroid polyhedra. Networks, 15(2), 205–215.

    Article  MATH  MathSciNet  Google Scholar 

  21. Cunningham, W. H. (1985). On submodular function minimization. Combinatorica, 5, 185–192.

    Article  MATH  MathSciNet  Google Scholar 

  22. Dechter, R. (2003). Constraint processing. Morgan Kaufmann.

  23. Feder, T., & Vardi, M. Y. (1998). 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.

    Article  MATH  MathSciNet  Google Scholar 

  24. Fujishige, S. (2005). Submodular Functions and Optimisation. 2nd edn., Annals of discrete mathematics 58. Elsevier.

  25. Fujishige, S., Hayashi, T., & Isotani, S. (2006). The minimum-norm-point algorithm applied to submodular function minimization and linear programming. Report RIMS1571, Research Institute for Mathematical Sciences, Kyoto University.

  26. Fujishige, S., & Patkar, S. B. (2001). Realization of set functions as cut functions of graphs and hypergraphs. Discrete Mathematics, 226, 199–210.

    Article  MATH  MathSciNet  Google Scholar 

  27. Goldberg, A., & Tarjan, R. E. (1988). A new approach to the maximum flow problem. Journal of the ACM, 35, 921–940.

    Article  MATH  MathSciNet  Google Scholar 

  28. Grötschel, M., Lovász, L., & Schrijver, A. (1981). The ellipsoid method and its consequences in combinatorial optimization. Combinatorica, 1, 169–198. (1984). Corrigendum: Combinatorica, 4, 291–295.

  29. Iwata, S. (2002). A fully combinatorial algorithm for submodular function minimization. Journal of Combinatorial Theory, Series B, 84(2), 203–212.

    Article  MATH  MathSciNet  Google Scholar 

  30. Iwata, S. (2002). A faster scaling algorithm for minimizing submodular functions. SIAM Journal on Computing, 32(4), 833–840.

    Article  MathSciNet  Google Scholar 

  31. Iwata, S., Fleischer, L., & Fujishige, S. (2001). A combinatorial, strongly polynomial-time algorithm for minimizing submodular functions. Journal of the ACM, 48(4), 761–777.

    Article  MATH  MathSciNet  Google Scholar 

  32. Jeavons, P.G. (1998). On the algebraic structure of combinatorial problems. Theoretical Computer Science, 200, 185–204.

    Article  MATH  MathSciNet  Google Scholar 

  33. Jeavons, P., Cohen, D., & Cooper M. C. (1998). Constraints, consistency and closure. Artificial Intelligence, 101, 251–265.

    Article  MATH  MathSciNet  Google Scholar 

  34. Jeavons, P. G., Cohen D. A., & Gyssens, M. (1997). Closure properties of constraints. Journal of the ACM, 44, 527–548.

    Article  MATH  MathSciNet  Google Scholar 

  35. Jeavons, P. G., & Cooper, M. C. (1995). Tractable constraints on ordered domains. Artificial Intelligence, 79(2), 327–339.

    Article  MATH  MathSciNet  Google Scholar 

  36. Jonsson, P., Klasson, M., & Krokhin, A. (2006). The approximability of three-valued Max CSP. SIAM Journal on Computing, 35(6), 1329–1349.

    Article  MATH  MathSciNet  Google Scholar 

  37. Koster, A. (1999). Frequency assignment—models and algorithms. PhD thesis, The Netherlands: Universiteit Maastricht, ISBN 90-9013119-1.

  38. Koster, A. M. C. A., van Hoesel, S. P. M., & Kolen, A. W. J. (1998). The partial constraint satisfaction problem: Facets and lifting theorems. Operations Research Letters, 23(3–5), 89–97.

    Article  MATH  MathSciNet  Google Scholar 

  39. Larrosa, J., & Schiex, T. (2004). Solving weighted CSP by maintaining arc consistency. Artificial Intelligence, 159, 1–26.

    Article  MATH  MathSciNet  Google Scholar 

  40. Lecoutre, C., & Szymanek, R. (2006). Generalized arc consistency for positive table constraints. In F. Benhamou (Ed.), Proc. principles and practice of constraint programming - CP 2006, LNCS 4204 (pp. 284–298). Berlin: Springer-Verlag.

    Chapter  Google Scholar 

  41. McCormick, S. T. (2005). Submodular function minimization in discrete optimization. In K. Aardal, G. L. Nemhauser, & R. Weismantel (Eds.), Handbooks in operations research and management science 12 (pp. 321–391). Amsterdam: Elsevier.

    Google Scholar 

  42. Meseguer, P., Rossi, F., & Schiex, T. (2006). Soft constraints. In F. Rossi, P. van Beek, & T. Walsh (Eds.), Handbook of constraint programming (pp. 281–328). Elsevier.

  43. Mohr, R., & Masini, G. (1988) Good old discrete relaxation. In Proc. European conf. artificial intelligence (pp. 651–656). Munich.

  44. Narayanan, H. (1997). Submodular functions and electrical networks. North-Holland, Amsterdam.

    MATH  Google Scholar 

  45. Orlin, J. B. (2007). A faster strongly polynomial time algorithm for submodular minimization. In M. Fischetti, & D. P. Williamson (Eds.), IPCO 2007, LNCS 4513 (pp. 240–251).

  46. Schiex, T., Fargier, H., & Verfaillie, G. (1995). Valued constraint satisfaction problems: Hard and easy problems. In Proc. of the 14th IJCAI (pp. 631–637). Montreal, Canada.

  47. Schlesinger, D. (2007). Exact solution of permuted submodular MinSum problems. In Proc. of the 6th international conference on energy minimization methods in computer vision and pattern recognition, Ezhou, Hubei, China LNCS 4679 (Vol. 4679, pp. 28–38) Springer.

  48. Schlesinger, M. (1976). Syntactic analysis of two-dimensional visual signals in noisy conditions. Kibernetika, 4, 113–130 (In Russian).

    Google Scholar 

  49. Schrijver, A. (2000). A combinatorial algorithm minimizing submodular functions in strongly polynomial time. Journal of Combinatorial Theory, Series B, 80, 346–355.

    Article  MATH  MathSciNet  Google Scholar 

  50. Topkis, D. (1998). Supermodularity and complementarity. Princeton University Press.

  51. Werner, T. (2007). A linear programming approach to max-sum problem: A review. IEEE Transactions on Pattern Analysis and Machine Intelligence, 29(7), 1165–1179.

    Article  Google Scholar 

  52. Wolfe, P. (1976). Finding the nearest point in a polytope. Mathematical Programming, 11, 128–149.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. IRIT, University of Toulouse III, 31062, Toulouse, France

    Martin C. Cooper

Authors
  1. Martin C. Cooper
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martin C. Cooper.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cooper, M.C. Minimization of Locally Defined Submodular Functions by Optimal Soft Arc Consistency. Constraints 13, 437–458 (2008). https://doi.org/10.1007/s10601-007-9037-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10601-007-9037-5

Keywords

Search

Navigation