The Complexity of Boolean Surjective General-Valued CSPs

Abstract

Valued constraint satisfaction problems (VCSPs) are discrete optimisation problems with a (Q ∪ {∞})-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, ∞}-valued constraint languages (corresponding to surjective decision CSPs) obtained by Creignou and Hébrard. For the maximisation problem of Q ≥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. * Extended abstracts of parts of this work appeared in

PTAS for Sparse General-Valued CSPs

“…This follows from the NP-hardness result of Kozik and Ochremiak[25], which actually shows APX-hardness; for earlier, explicit APX-hardness results for CSPs see, e.g.,[4],[26]. However, we remark non-trivial PTAS examples are known for "surjective" maximisation finite-valued CSPs[27].…”

Algebraic Global Gadgetry for Surjective Constraint Satisfaction

“…years (Creignou et al 2001, Theorem 6.12), but a classification for all 3-element structures is currently open. Here, we can mention that optimization and counting flavors of the surjective CSP have also been studied (Chen et al 2019;Focke et al 2019;Fulla et al 2019).…”