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Non-expressibility of Submodular Languages

  • Chapter
The Complexity of Valued Constraint Satisfaction Problems

Part of the book series: Cognitive Technologies ((COGTECH))

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Abstract

Chapter 5 studies the problem of whether all Boolean submodular functions can be decomposed into a sum of binary submodular functions over a possibly larger set of variables. This problem has been considered within several different contexts in computer science, including computer vision, artificial intelligence, and pseudo-Boolean optimisation. Using the connection between the expressive power of submodular functions and certain algebraic properties of functions described in earlier chapters, a negative answer to this question is shown. Consequently, there are submodular functions that cannot be reduced to the minimum cut problem via the expressibility reduction.

A man should look for what is, and not for what he thinks should be.

Albert Einstein

This chapter is reprinted from Discrete Applied Mathematics, 157(15), S. Živný, D.A. Cohen, and P.G. Jeavons, The Expressive Power of Binary Submodular Functions, 3347–3358, Copyright (2009), with permission from Elsevier.

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Notes

  1. 1.

    For example, standard combinatorial counting techniques cannot resolve this question because we allow arbitrary real-valued coefficients in submodular polynomials. We also allow an arbitrary number of additional variables.

  2. 2.

    As implemented by the program Skeleton available from http://www.uic.nnov.ru/~zny/skeleton/.

  3. 3.

    Optimal (in the number of extra variables) gadgets for all cost functions from Γ fans,4 have been identified in [278].

References

  1. Besag, J.: On the statistical analysis of dirty pictures. J. R. Stat. Soc. B 48(3), 259–302 (1986)

    MathSciNet  MATH  Google Scholar 

  2. Billionet, A., Minoux, M.: Maximizing a supermodular pseudo-Boolean function: a polynomial algorithm for cubic functions. Discrete Appl. Math. 12(1), 1–11 (1985)

    Article  MathSciNet  Google Scholar 

  3. Boros, E., Gruber, A.: On quadratization of pseudo-Boolean functions. In: Proceedings of the International Symposium on Artificial Intelligence and Mathematics (ISAIM’12) (2012)

    Google Scholar 

  4. Charpiat, G.: Exhaustive family of energies minimizable exactly by a graph cut. In: Proceedings of the 24th IEEE Conference on Computer Vision and Pattern Recognition (CVPR’11), pp. 1849–1856. IEEE, New York (2011)

    Google Scholar 

  5. Cohen, D., Cooper, M., Jeavons, P., Krokhin, A.: A maximal tractable class of soft constraints. J. Artif. Intell. Res. 22, 1–22 (2004)

    MathSciNet  MATH  Google Scholar 

  6. Cohen, D., Cooper, M., Jeavons, P., Krokhin, A.: Supermodular functions and the complexity of MAX-CSP. Discrete Appl. Math. 149(1–3), 53–72 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cohen, D.A., Cooper, M.C., Jeavons, P.G.: An algebraic characterisation of complexity for valued constraints. In: Proceedings of the 12th International Conference on Principles and Practice of Constraint Programming (CP’06). Lecture Notes in Computer Science, vol. 4204, pp. 107–121. Springer, Berlin (2006)

    Google Scholar 

  8. Cohen, D.A., Creed, P., Jeavons, P.G., Živný, S.: An algebraic theory of complexity for valued constraints: establishing a Galois connection. In: Proceedings of the 36th International Symposium on Mathematical Foundations of Computer Science (MFCS’11). Lecture Notes in Computer Science, vol. 6907, pp. 231–242. Springer, Berlin (2011)

    Google Scholar 

  9. Crama, Y.: Recognition problems for special classes of polynomials in 0–1 variables. Math. Program. 44(1–3), 139–155 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  10. Creignou, N., Khanna, S., Sudan, M.: Complexity Classification of Boolean Constraint Satisfaction Problems. SIAM Monographs on Discrete Mathematics and Applications, vol. 7. SIAM, Philadelphia (2001)

    Book  Google Scholar 

  11. Dechter, R., Pearl, J.: Tree clustering for constraint networks. Artif. Intell. 38(3), 353–366 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fujishige, S., Patkar, S.B.: Realization of set functions as cut functions of graphs and hypergraphs. Discrete Math. 226(1–3), 199–210 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gallo, G., Simeone, B.: On the supermodular knapsack problem. Math. Program. 45(1–3), 295–309 (1988)

    MathSciNet  Google Scholar 

  14. Geman, S., Geman, D.: Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell. 6(6), 721–741 (1984)

    Article  MATH  Google Scholar 

  15. Hansen, P., Simeone, B.: Unimodular functions. Discrete Appl. Math. 14(3), 269–281 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  16. Iwata, S., Orlin, J.B.: A simple combinatorial algorithm for submodular function minimization. In: Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’09), pp. 1230–1237. SIAM, Philadelphia (2009)

    Google Scholar 

  17. Kahl, F., Strandmark, P.: Generalized roof duality for pseudo-Boolean optimization. In: Proceedings of the 13th IEEE International Conference on Computer Vision (ICCV’11), pp. 255–262. IEEE, New York (2011)

    Google Scholar 

  18. Kohli, P., Kumar, M.P., Torr, P.H.S.: P3 & beyond: solving energies with higher order cliques. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’07). IEEE Computer Society, Los Alamitos (2007)

    Google Scholar 

  19. Kolmogorov, V.: Minimizing a sum of submodular functions. Discrete Appl. Math. 160(15), 2246–2258 (2012)

    Article  MathSciNet  Google Scholar 

  20. Lan, X., Roth, S., Huttenlocher, D.P., Black, M.J.: Efficient belief propagation with learned higher-order Markov random fields. In: Proceedings of the 9th European Conference on Computer Vision (ECCV’06), Part II. Lecture Notes in Computer Science, vol. 3952, pp. 269–282. Springer, Berlin (2006)

    Google Scholar 

  21. Larrosa, J., Dechter, R.: On the dual representation of non-binary semiring-based CSPs. In: Workshop on Soft Constraints (CP’00) (2000)

    Google Scholar 

  22. Lauritzen, S.L.: Graphical Models. Oxford University Press, London (1996)

    Google Scholar 

  23. Motzkin, T., Raiffa, H., Thompson, G., Thrall, R.: The double description method. In: Kuhn, H.W., Tucker, A.W. (eds.) Contributions to the Theory of Games, vol. 2, pp. 51–73. Princeton University Press, Princeton (1953)

    Google Scholar 

  24. Paget, R., Longstaff, I.D.: Texture synthesis via a noncausal nonparametric multiscale Markov random field. IEEE Trans. Image Process. 7(6), 925–931 (1998)

    Article  Google Scholar 

  25. Promislow, S., Young, V.: Supermodular functions on finite lattices. Order 22(4), 389–413 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  26. Rhys, J.: A selection problem of shared fixed costs and network flows. Manag. Sci. 17(3), 200–207 (1970)

    Article  MATH  Google Scholar 

  27. Roth, S., Black, M.J.: Fields of experts: a framework for learning image priors. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’05), pp. 860–867. IEEE Computer Society, Los Alamitos (2005)

    Chapter  Google Scholar 

  28. Rother, C., Kohli, P., Feng, W., Jia, J.: Minimizing sparse higher order energy functions of discrete variables. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’09). IEEE Computer Society, Los Alamitos (2009)

    Google Scholar 

  29. Wainwright, M.J., Jordan, M.I.: Graphical models, exponential families, and variational inference. Found. Trends Mach. Learn. 1(1–2), 1–305 (2008)

    MATH  Google Scholar 

  30. Werner, T.: A linear programming approach to Max-Sum problem: a review. IEEE Trans. Pattern Anal. Mach. Intell. 29(7), 1165–1179 (2007)

    Article  Google Scholar 

  31. Živný, S., Cohen, D.A., Jeavons, P.G.: The expressive power of binary submodular functions. Discrete Appl. Math. 157(15), 3347–3358 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Živný, S., Jeavons, P.G.: Which submodular functions are expressible using binary submodular functions? Research Report cs-rr-08-08, Computing Laboratory, University of Oxford, Oxford, UK (2008)

    Google Scholar 

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  1. University College, University of Oxford, Oxford, UK

    Stanislav Živný

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  1. Stanislav Živný
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Živný, S. (2012). Non-expressibility of Submodular Languages. In: The Complexity of Valued Constraint Satisfaction Problems. Cognitive Technologies. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33974-5_5

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  • DOI: https://doi.org/10.1007/978-3-642-33974-5_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-33973-8

  • Online ISBN: 978-3-642-33974-5

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