The Complexity of General-Valued Constraint Satisfaction Problems Seen from the Other Side
Abstract
The constraint satisfaction problem (CSP) is concerned with homomorphisms between two structures. For CSPs with restricted left-hand-side structures, the results of Dalmau, Kolaitis, and Vardi [Proceedings of the 8th International Conference on Principles and Practice of Constraint Programming, Springer, New York, 2002, pp. 310--326], Grohe [J. ACM, 54 (2007), 1], and Atserias, Bulatov, and Dalmau [Proceedings of the 34th International Colloquium on Automata, Languages and Programming, Springer, New York, 2007, pp. 279--290] establish the precise borderline of polynomial-time solvability (subject to complexity-theoretic assumptions) and of solvability by bounded-consistency algorithms (unconditionally) as bounded treewidth modulo homomorphic equivalence. The general-valued constraint satisfaction problem (VCSP) is a generalization of the CSP concerned with homomorphisms between two valued structures. For VCSPs with restricted left-hand-side valued structures, we establish the precise borderline of polynomial-time solvability (subject to complexity-theoretic assumptions) and of solvability by the $k$th level of the Sherali--Adams LP hierarchy (unconditionally). We also obtain results on related problems concerned with finding a solution and recognizing the tractable cases; the latter has an application in database theory.
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Submitted: 15 March 2019
Accepted: 13 October 2021
Published online: 25 January 2022
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Horizon 2020 Framework Programme https://doi.org/10.13039/100010661
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Fondo Nacional de Desarrollo Científico y Tecnológico https://doi.org/10.13039/501100002850
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Royal Society https://doi.org/10.13039/501100000288
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