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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 4 focuses on the expressive power of binary submodular functions. Known and new classes of submodular functions that are expressible by binary submodular functions are presented. There is a close relationship between the expressive power of binary submodular functions and solving submodular valued constraint satisfaction problems via the minimum cut problem: showing that a class \(\mathcal{C}\) of submodular functions is expressible by binary submodular functions is equivalent to showing that functions from \(\mathcal{C}\) can be minimised efficiently via the minimum cut problem.

An algorithm must be seen to be believed.

Donald Knuth

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Notes

  1. 1.

    In fact, it is known that all Boolean polynomials (of arbitrary degree) are expressible by binary polynomials [34, 247], but the general construction does not preserve submodularity, that is, the resulting binary polynomials are not necessarily submodular.

  2. 2.

    Note that this result is not obvious because simply changing the infinite cost to some big, but finite constant M does not work: for c 1<c 2, ∞+c 1≥∞+c 2, but M+c 1<M+c 2. For instance, consider the submodular cost function ϕ defined as follows: ϕ(0,0)=ϕ(1,0)=∞, ϕ(0,1)=1, and ϕ(1,1)=2. Changing ϕ(0,0)=ϕ(1,0)=M for any finite number M would violate the submodularity condition.

  3. 3.

    In [59], 2-monotone cost functions are defined over lattice-ordered sets and called generalised 2-monotone cost functions. In [191], these are called just 2-monotone cost functions.

  4. 4.

    Most papers on pseudo-Boolean optimisation deal with the maximisation problem, and therefore talk about supermodular functions, rather than submodular functions. However, the maximisation problem of supermodular functions is equivalent to the minimisation problem of submodular functions.

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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). 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_4

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  • DOI: https://doi.org/10.1007/978-3-642-33974-5_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-33973-8

  • Online ISBN: 978-3-642-33974-5

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