On The Power of Tree Projections: Structural Tractability of Enumerating CSP Solutions

The problem of deciding whether CSP instances admit solutions has been deeply studied in the literature, and several structural tractability results have been derived so far. However, constraint satisfaction comes in practice as a computation problem where the focus is either on finding one solution, or on enumerating all solutions, possibly projected to some given set of output variables. The paper investigates the structural tractability of the problem of enumerating (possibly projected) solutions, where tractability means here computable with polynomial delay (WPD), since in general exponentially many solutions may be computed. A general framework based on the notion of tree projection of hypergraphs is considered, which generalizes all known decomposition methods. Tractability results have been obtained both for classes of structures where output variables are part of their specification, and for classes of structures where computability WPD must be ensured for any possible set of output variables. These results are shown to be tight, by exhibiting dichotomies for classes of structures having bounded arity and where the tree decomposition method is considered

An Algebraic Approach to Valued Constraint Satisfaction

[EN]We study the complexity of the valued CSP (VCSP, for short) over arbitrary templates, taking the general framework of integral bounded linearly order monoids as valuation structures. The class of problems considered here subsumes and generalizes the most common one in VCSP literature, since both monoidal and lattice conjunction operations are allowed in the formulation of constraints. Restricting to locally finite monoids, we introduce a notion of polymorphism that captures the pp-definability in the style of Geiger’s result. As a consequence, sufficient conditions for tractability of the classical CSP, related to the existence of certain polymorphisms, are shown to serve also for the valued case. Finally, we establish the dichotomy conjecture for the VCSP, modulo the dichotomy for classical CSP.The work was partly supported by the grant No. GA17-04630S of the Czech Science Foundation and partly by the long-term strategic development financing of the Institute of Computer Science (RVO:67985807).Peer reviewe

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