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The Power of Linear Programming for Valued CSPs
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. An instance of the problem is specified by a sum of cost 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. Using this result, 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.Comment: Corrected a few typo
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