The complexity of conservative valued CSPs

Abstract

We study the complexity of valued constraint satisfaction problems (VCSP). A problem from VCSP is characterised by a constraint language, a fixed set of cost functions over a finite domain. An instance of the problem is specified by a sum of cost functions from the language and the goal is to minimise the sum. Under the unique games conjecture, the approximability of finite-valued VCSPs is wellunderstood, see Raghavendra [FOCS'08]. However, there is no characterisation of finite-valued VCSPs, let alone general-valued VCSPs, that can be solved exactly in polynomial time, thus giving insights from a combinatorial optimisation perspective.We consider the case of languages containing all possible unary cost functions. We prove a Schaefer-like dichotomy theorem for conservative valued languages: if all cost functions in the language satisfy a certain condition (specified by a complementary combination of STP and MJN multimorphisms), then any instance can be solved in polynomial time (via a new algorithm developed in this paper), otherwise the language is NP-hard. This is the first complete complexity classification of general-valued constraint languages over non-Boolean domains. It is a common phenomenon that complexity classifications of problems over non-Boolean domains is significantly harder than the Boolean case. The polynomial-time algorithm we present for the tractable cases is a generalisation of the submodular minimisation problem and a result of Cohen et al. [TCS'08].Our results generalise previous results by Takhanov [STACS'10] and (a subset of results) by Cohen et al. [AIJ'06] and Deineko et al. [JACM'08]. Moreover, our results do not rely on any computerassisted search as in Deineko et al. [JACM'08], and provide a powerful tool for proving hardness of finite-valued and general-valued languages.

Effectiveness of Structural Restrictions for Hybrid CSPs

“…finitevalued languages [35]. Informally, the difficulty can be attributed to the fact that tractable finite-valued languages are characterized by fractional polymorphisms with an arbitrarily large support (if the size of the domain is not fixed), whereas for conservative languages we need two fractional polymorphisms that contain a constant number of operations in the support, namely 2 and 3 [24].…”

Algebraic Properties of Valued Constraint Satisfaction Problem

“…By [19] if follows from Claims 2 and 3 that Γ admits a conservative ternary multimorphism Mj 1 , Mj 2 , Mn 3 such that for every {x, y} ∈ M , Mj 1 , Mj 2 , Mn 3 restricted to {x, y} is an MJN, which finishes the proof of Proposition 42.…”

“…The proof of the other implication consists of a sequence of claims and heavily relies on the results of [19].…”

A complexity classification of spin systems with an external field

“…In ref. 1 it is shown that distinguishing between the possibilities reduces to deciding whether a certain finite system (based on A) has a property called "balance"-a problem that is known (5) to be decidableand to deciding whether A has an algebraic invariant called an STP-MJN multimorphism (6). This can be checked by brute force, given A.…”