The complexity of Boolean surjective general-valued CSPs

Valued constraint satisfaction problems (VCSPs) are discrete optimisation problems with a $(\mathbb{Q}\cup\{\infty\})$-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 $\{0,\infty\}$-valued constraint languages (corresponding to surjective decision CSPs) obtained by Creignou and H\'ebrard. For the maximisation problem of $\mathbb{Q}_{\geq 0}$-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.Comment: v5: small corrections and improved presentatio

PTAS for Sparse General-Valued CSPs

We study polynomial-time approximation schemes (PTASes) for constraint satisfaction problems (CSPs) such as Maximum Independent Set or Minimum Vertex Cover on sparse graph classes. Baker's approach gives a PTAS on planar graphs, excluded-minor classes, and beyond. For Max-CSPs, and even more generally, maximisation finite-valued CSPs (where constraints are arbitrary non-negative functions), Romero, Wrochna, and \v{Z}ivn\'y [SODA'21] showed that the Sherali-Adams LP relaxation gives a simple PTAS for all fractionally-treewidth-fragile classes, which is the most general "sparsity" condition for which a PTAS is known. We extend these results to general-valued CSPs, which include "crisp" (or "strict") constraints that have to be satisfied by every feasible assignment. The only condition on the crisp constraints is that their domain contains an element which is at least as feasible as all the others (but possibly less valuable). For minimisation general-valued CSPs with crisp constraints, we present a PTAS for all Baker graph classes -- a definition by Dvo\v{r}\'ak [SODA'20] which encompasses all classes where Baker's technique is known to work, except possibly for fractionally-treewidth-fragile classes. While this is standard for problems satisfying a certain monotonicity condition on crisp constraints, we show this can be relaxed to diagonalisability -- a property of relational structures connected to logics, statistical physics, and random CSPs

The complexity of Boolean surjective general-valued CSPs

Valued constraint satisfaction problems (VCSPs) are discrete optimisation problems with a $\overline{\mathbb{Q}}$-valued objective function given as a sum of fixed-arity functions, where $\overline{\mathbb{Q}}=\mathbb{Q}\cup\{\infty\}$ is the set of extended rationals. In Boolean surjective VCSPs variables take on labels from $D=\{0,1\}$ and an optimal assignment is required to use both labels from $D$. A classic example is the global min-cut problem in graphs. Building on the work of Uppman, we establish a dichotomy theorem and thus give a complete complexity classification of Boolean surjective VCSPs. The newly discovered tractable case has an interesting structure related to projections of downsets and upsets. Our work generalises the dichotomy for $\{0,\infty\}$-valued constraint languages (corresponding to CSPs) obtained by Creignou and H\&amp;apos;ebrard, and the dichotomy for $\{0,1\}$-valued constraint languages (corresponding to Min-CSPs) obtained by Uppman.</p

The complexity of Boolean surjective general-valued CSPs

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 H´ebrard. 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