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The Complexity of Valued Constraint Satisfaction Problems pp 131–135Cite as

The Power of Linear Programming

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

Abstract

Chapter 8 shows recent results on the power of linear programming for valued constraint satisfaction problems. In particular, it presents an algebraic characterisation, in terms of fractional polymorphisms, of valued constraint languages that are tractable via a standard linear programming relaxation. This result has several algorithmic consequences such as establishing the tractability of submodular functions on trees.

You don’t understand anything until you learn it more than one way.

Marvin Minsky

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Notes

  1. 1.

    A similar structure for {0,∞}-valued languages was introduced in [197].

  2. 2.

    The algorithm from [118] is sometimes called the generalised arc consistency algorithm to emphasise the fact that it works for CSPs of arbitrary arities, and not only for binary CSPs [207].

  3. 3.

    A semi-lattice operation is associative, commutative, and idempotent.

  4. 4.

    Namely, the STP might contain cycles, but [66, Lemma 7.15] tells us that on cycles we have, in the finite-valued case, only unary cost functions. It follows that the cost functions admitting the STP must be submodular with respect to some total order.

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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). The Power of Linear Programming. In: The Complexity of Valued Constraint Satisfaction Problems. Cognitive Technologies. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33974-5_8

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  • DOI: https://doi.org/10.1007/978-3-642-33974-5_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-33973-8

  • Online ISBN: 978-3-642-33974-5

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