The complexity of conservative valued CSPs
Journal of the ACM (JACM)
The complexity of finite-valued CSPs
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
The power of linear programming for finite-valued CSPs: a constructive characterization
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
The complexity of three-element min-sol and conservative min-cost-hom
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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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) sub modular on arbitrary lattices, (2) bisubmodular (also known as k-sub modular) on arbitrary finite domains, (3) weakly (and hence strongly) tree-sub modular on arbitrary trees.