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The Complexity of Valued Constraint Satisfaction Problems

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

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Abstract

Chapter 1 introduces the necessary background. It defines valued constraint satisfaction problems and gives examples of problems that can be formulated as such. Secondly, it defines the key concept of expressibility and introduces certain algebraic properties that play a key role in the complexity analysis of valued constraint satisfaction problems. Finally, it presents the notion of submodularity and surveys known results regarding optimising submodular functions.

The highest technique is to have no technique.

Bruce Lee

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Notes

  1. 1.

    A valuation structure, Ω, is a totally-ordered set, with a minimum and a maximum element (denoted by 0 and by ∞), together with a commutative, associative binary aggregation operator, ⊕, such that for all α,β,γ∈Ω, α⊕0=α and α⊕γ≥β⊕γ whenever α≥β.

  2. 2.

    The main difference between semi-ring CSPs and valued CSPs is that costs in valued CSPs represent violation levels and have to be totally ordered, whereas costs in semi-ring CSPs represent preferences and may be ordered only partially.

  3. 3.

    This is true for optimal solutions. However, if we are interested in approximability results, this statement is not true even over Boolean domains; see [87].

  4. 4.

    The power of the local consistency technique has also been fully characterised for uniform CSPs [4].

  5. 5.

    Polymorphisms of valued constraint languages are also known as feasibility polymorphisms [65].

  6. 6.

    For a k-ary fractional operation, all weights r i are divided by k.

  7. 7.

    Think of 2V as {0,1}-vectors of length |V|.

  8. 8.

    Let \(\mathcal{B}\) be an algorithm for the minimisation problem of f in the oracle value model. \(\mathcal{B}\) is called polynomial if it runs in polynomial time. A polynomial algorithm \(\mathcal{B}\) is called strongly polynomial if the running time does not depend on M=maxf. In other words, the number of elementary arithmetic operations and other operations is bounded by a polynomial in the size of the input. A polynomial algorithm \(\mathcal{B}\) which does depend on M is called weakly polynomial. A polynomial algorithm \(\mathcal{B}\) is called combinatorial if it does not employ the ellipsoid method. Finally, a combinatorial algorithm \(\mathcal{B}\) is called fully combinatorial if it uses only oracle calls, additions, subtractions, and comparisons, but not multiplications and divisions, as fundamental operations.

  9. 9.

    The class of cost functions closed under a tournament pair multimorphism is more general than the class of submodular cost functions if the range of the cost functions includes infinite costs [66].

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    Stanislav Živný

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Živný, S. (2012). Background. In: The Complexity of Valued Constraint Satisfaction Problems. Cognitive Technologies. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33974-5_1

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