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.
Keywords
- Submodular Cost Functions
- Pseudo-boolean Optimization
- Multimorphism
- Fractional Polymorphisms
- Higher Order Energy Functions
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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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- 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.
As implemented by the program Skeleton available from http://www.uic.nnov.ru/~zny/skeleton/.
- 3.
Optimal (in the number of extra variables) gadgets for all cost functions from Γ fans,4 have been identified in [278].
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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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