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  • 1.
    Fulla, Peter
    et al.
    Univ Oxford, England.
    Uppman, Hannes
    Linköping University, Department of Computer and Information Science, Software and Systems. Linköping University, Faculty of Science & Engineering.
    Zivny, Stanislav
    Univ Oxford, England.
    The Complexity of Boolean Surjective General-Valued CSPs2019In: ACM Transactions on Computation Theory, ISSN 1942-3454, E-ISSN 1942-3462, Vol. 11, no 1, article id 4Article in journal (Refereed)
    Abstract [en]

    Valued constraint satisfaction problems (VCSPs) are discrete optimisation problems with a (Q boolean OR{infinity})-valued objective function given as a sum of fixed-arity functions. In Boolean surjective VCSPs, variables take on labels from D = {0, 1}, and an optimal assignment is required to use both labels from D. Examples include the classical global Min-Cut problem in graphs and the Minimum Distance problem studied in coding theory. We establish a dichotomy theorem and thus give a complete complexity classification of Boolean surjective VCSPs with respect to exact solvability. Our work generalises the dichotomy for {0, infinity}-valued constraint languages (corresponding to surjective decision CSPs) obtained by Creignou and Hebrard. For the maximisation problem of Q(amp;gt;= 0)-valued surjective VCSPs, we also establish a dichotomy theorem with respect to approximability. Unlike in the case of Boolean surjective (decision) CSPs, there appears a novel tractable class of languages that is trivial in the non-surjective setting. This newly discovered tractable class has an interesting mathematical structure related to downsets and upsets. Our main contribution is identifying this class and proving that it lies on the borderline of tractability. A crucial part of our proof is a polynomial-time algorithm for enumerating all near-optimal solutions to a generalised Min-Cut problem, which might be of independent interest.

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