Editors: Valentin Goranko and Mads Dam; Article No. 42

Rostislav Horčík, Tommaso Moraschini, Amanda Vidal
20 Leibniz International Proceedings in Informatics Schloss Dagstuhl-Leibniz-Zentrum für Informatik   unpublished
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
more » ... he 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. 1 Introduction The constraint satisfaction problem (CSP, for short) is a well-established framework for the uniform study of a wide range of both theoretical and applied problems. For the present purpose, it will be convenient to describe it in purely logical terms. A first-order sentence is primitive positive (pp, for short) if it is built up from atomic formulas with equality using only conjunction and existential quantifier. The CSP of a finite relational structure B asks to determine the pp-sentences valid in B, in symbols CSP(B) := {ϕ | ϕ is pp-sentence and B ϕ}. It is well known that CSP(B) can be identified with the set of finite structures A for which there is a homomorphism A → B.
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