The power of linear programming for general-valued CSPs

V. Kolmogorov, J. Thapper, S. Živný, SIAM Journal on Computing 44 (2015) 1–36.


Journal Article | Published | English
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Abstract
A class of valued constraint satisfaction problems (VCSPs) is characterised by a valued constraint language, a fixed set of cost functions on a finite domain. Finite-valued constraint languages contain functions that take on rational costs and general-valued constraint languages contain functions that take on rational or infinite costs. An instance of the problem is specified by a sum of functions from the language with the goal to minimise the sum. This framework includes and generalises well-studied constraint satisfaction problems (CSPs) and maximum constraint satisfaction problems (Max-CSPs). Our main result is a precise algebraic characterisation of valued constraint languages whose instances can be solved exactly by the basic linear programming relaxation (BLP). For a general-valued constraint language Γ, BLP is a decision procedure for Γ if and only if Γ admits a symmetric fractional polymorphism of every arity. For a finite-valued constraint language Γ, BLP is a decision procedure if and only if Γ admits a symmetric fractional polymorphism of some arity, or equivalently, if Γ admits a symmetric fractional polymorphism of arity 2. Using these results, we obtain tractability of several novel and previously widely-open classes of VCSPs, including problems over valued constraint languages that are: (1) submodular on arbitrary lattices; (2) bisubmodular (also known as k-submodular) on arbitrary finite domains; (3) weakly (and hence strongly) tree-submodular on arbitrary trees.
Publishing Year
Date Published
2015-02-01
Journal Title
SIAM Journal on Computing
Volume
44
Issue
1
Page
1 - 36
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Cite this

Kolmogorov V, Thapper J, Živný S. The power of linear programming for general-valued CSPs. SIAM Journal on Computing. 2015;44(1):1-36. doi:10.1137/130945648
Kolmogorov, V., Thapper, J., & Živný, S. (2015). The power of linear programming for general-valued CSPs. SIAM Journal on Computing, 44(1), 1–36. https://doi.org/10.1137/130945648
Kolmogorov, Vladimir, Johan Thapper, and Stanislav Živný. “The Power of Linear Programming for General-Valued CSPs.” SIAM Journal on Computing 44, no. 1 (2015): 1–36. https://doi.org/10.1137/130945648.
V. Kolmogorov, J. Thapper, and S. Živný, “The power of linear programming for general-valued CSPs,” SIAM Journal on Computing, vol. 44, no. 1, pp. 1–36, 2015.
Kolmogorov V, Thapper J, Živný S. 2015. The power of linear programming for general-valued CSPs. SIAM Journal on Computing. 44(1), 1–36.
Kolmogorov, Vladimir, et al. “The Power of Linear Programming for General-Valued CSPs.” SIAM Journal on Computing, vol. 44, no. 1, SIAM, 2015, pp. 1–36, doi:10.1137/130945648.

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