1,886 research outputs found

    The complexity of Boolean surjective general-valued CSPs

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    Valued constraint satisfaction problems (VCSPs) are discrete optimisation problems with a (Q∪{∞})(\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}D=\{0,1\} and an optimal assignment is required to use both labels from DD. 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,∞}\{0,\infty\}-valued constraint languages (corresponding to surjective decision CSPs) obtained by Creignou and H\'ebrard. For the maximisation problem of Q≥0\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

    The Complexity of General-Valued CSPs

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    An instance of the Valued Constraint Satisfaction Problem (VCSP) is given by a finite set of variables, a finite domain of labels, and a sum of functions, each function depending on a subset of the variables. Each function can take finite values specifying costs of assignments of labels to its variables or the infinite value, which indicates an infeasible assignment. The goal is to find an assignment of labels to the variables that minimizes the sum. We study, assuming that P ≠ NP, how the complexity of this very general problem depends on the set of functions allowed in the instances, the so-called constraint language. The case when all allowed functions take values in {0, ∞} corresponds to ordinary CSPs, where one deals only with the feasibility issue and there is no optimization. This case is the subject of the Algebraic CSP Dichotomy Conjecture predicting for which constraint languages CSPs are tractable (i.e. solvable in polynomial time) and for which NP-hard. The case when all allowed functions take only finite values corresponds to finite-valued CSP, where the feasibility aspect is trivial and one deals only with the optimization issue. The complexity of finite-valued CSPs was fully classified by Thapper and Zivny. An algebraic necessary condition for tractability of a general-valued CSP with a fixed constraint language was recently given by Kozik and Ochremiak. As our main result, we prove that if a constraint language satisfies this algebraic necessary condition, and the feasibility CSP (i.e. the problem of deciding whether a given instance has a feasible solution) corresponding to the VCSP with this language is tractable, then the VCSP is tractable. The algorithm is a simple combination of the assumed algorithm for the feasibility CSP and the standard LP relaxation. As a corollary, we obtain that a dichotomy for ordinary CSPs would imply a dichotomy for general-valued CSPs

    The complexity of general-valued CSPs seen from the other side

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    The constraint satisfaction problem (CSP) is concerned with homomorphisms between two structures. For CSPs with restricted left-hand side structures, the results of Dalmau, Kolaitis, and Vardi [CP'02], Grohe [FOCS'03/JACM'07], and Atserias, Bulatov, and Dalmau [ICALP'07] establish the precise borderline of polynomial-time solvability (subject to complexity-theoretic assumptions) and of solvability by bounded-consistency algorithms (unconditionally) as bounded treewidth modulo homomorphic equivalence. The general-valued constraint satisfaction problem (VCSP) is a generalisation of the CSP concerned with homomorphisms between two valued structures. For VCSPs with restricted left-hand side valued structures, we establish the precise borderline of polynomial-time solvability (subject to complexity-theoretic assumptions) and of solvability by the kk-th level of the Sherali-Adams LP hierarchy (unconditionally). We also obtain results on related problems concerned with finding a solution and recognising the tractable cases; the latter has an application in database theory.Comment: v2: Full version of a FOCS'18 paper; improved presentation and small correction

    A Galois Connection for Weighted (Relational) Clones of Infinite Size

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    A Galois connection between clones and relational clones on a fixed finite domain is one of the cornerstones of the so-called algebraic approach to the computational complexity of non-uniform Constraint Satisfaction Problems (CSPs). Cohen et al. established a Galois connection between finitely-generated weighted clones and finitely-generated weighted relational clones [SICOMP'13], and asked whether this connection holds in general. We answer this question in the affirmative for weighted (relational) clones with real weights and show that the complexity of the corresponding valued CSPs is preserved

    Computational Complexity of the Minimum Cost Homomorphism Problem on Three-Element Domains

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    In this paper we study the computational complexity of the (extended) minimum cost homomorphism problem (Min-Cost-Hom) as a function of a constraint language, i.e. a set of constraint relations and cost functions that are allowed to appear in instances. A wide range of natural combinatorial optimisation problems can be expressed as Min-Cost-Homs and a classification of their complexity would be highly desirable, both from a direct, applied point of view as well as from a theoretical perspective. Min-Cost-Hom can be understood either as a flexible optimisation version of the constraint satisfaction problem (CSP) or a restriction of the (general-valued) valued constraint satisfaction problem (VCSP). Other optimisation versions of CSPs such as the minimum solution problem (Min-Sol) and the minimum ones problem (Min-Ones) are special cases of Min-Cost-Hom. The study of VCSPs has recently seen remarkable progress. A complete classification for the complexity of finite-valued languages on arbitrary finite domains has been obtained Thapper and Zivny [STOC'13]. However, understanding the complexity of languages that are not finite-valued appears to be more difficult. Min-Cost-Hom allows us to study problematic languages of this type without having to deal with with the full generality of the VCSP. A recent classification for the complexity of three-element Min-Sol, Uppman [ICALP'13], takes a step in this direction. In this paper we extend this result considerably by determining the complexity of three-element Min-Cost-Hom

    Hybrid VCSPs with crisp and conservative valued templates

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    A constraint satisfaction problem (CSP) is a problem of computing a homomorphism R→Γ{\bf R} \rightarrow {\bf \Gamma} between two relational structures. Analyzing its complexity has been a very fruitful research direction, especially for fixed template CSPs, denoted CSP(Γ)CSP({\bf \Gamma}), in which the right side structure Γ{\bf \Gamma} is fixed and the left side structure R{\bf R} is unconstrained. Recently, the hybrid setting, written CSPH(Γ)CSP_{\mathcal{H}}({\bf \Gamma}), where both sides are restricted simultaneously, attracted some attention. It assumes that R{\bf R} is taken from a class of relational structures H\mathcal{H} that additionally is closed under inverse homomorphisms. The last property allows to exploit algebraic tools that have been developed for fixed template CSPs. The key concept that connects hybrid CSPs with fixed-template CSPs is the so called "lifted language". Namely, this is a constraint language ΓR{\bf \Gamma}_{{\bf R}} that can be constructed from an input R{\bf R}. The tractability of that language for any input R∈H{\bf R}\in\mathcal{H} is a necessary condition for the tractability of the hybrid problem. In the first part we investigate templates Γ{\bf \Gamma} for which the latter condition is not only necessary, but also is sufficient. We call such templates Γ{\bf \Gamma} widely tractable. For this purpose, we construct from Γ{\bf \Gamma} a new finite relational structure Γ′{\bf \Gamma}' and define H0\mathcal{H}_0 as a class of structures homomorphic to Γ′{\bf \Gamma}'. We prove that wide tractability is equivalent to the tractability of CSPH0(Γ)CSP_{\mathcal{H}_0}({\bf \Gamma}). Our proof is based on the key observation that R{\bf R} is homomorphic to Γ′{\bf \Gamma}' if and only if the core of ΓR{\bf \Gamma}_{{\bf R}} is preserved by a Siggers polymorphism. Analogous result is shown for valued conservative CSPs.Comment: 21 pages. arXiv admin note: text overlap with arXiv:1504.0706

    The Power of Linear Programming for Valued CSPs

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    A class of valued constraint satisfaction problems (VCSPs) is characterised by a valued constraint language, a fixed set of cost functions on a finite domain. An instance of the problem is specified by a sum of cost functions from the language with the goal to minimise the sum. This framework includes and generalises well-studied constraint satisfaction problems (CSPs) and maximum constraint satisfaction problems (Max-CSPs). Our main result is a precise algebraic characterisation of valued constraint languages whose instances can be solved exactly by the basic linear programming relaxation. Using this result, we obtain tractability of several novel and previously widely-open classes of VCSPs, including problems over valued constraint languages that are: (1) submodular on arbitrary lattices; (2) bisubmodular (also known as k-submodular) on arbitrary finite domains; (3) weakly (and hence strongly) tree-submodular on arbitrary trees.Comment: Corrected a few typo

    Tractable Optimization Problems through Hypergraph-Based Structural Restrictions

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    Several variants of the Constraint Satisfaction Problem have been proposed and investigated in the literature for modelling those scenarios where solutions are associated with some given costs. Within these frameworks computing an optimal solution is an NP-hard problem in general; yet, when restricted over classes of instances whose constraint interactions can be modelled via (nearly-)acyclic graphs, this problem is known to be solvable in polynomial time. In this paper, larger classes of tractable instances are singled out, by discussing solution approaches based on exploiting hypergraph acyclicity and, more generally, structural decomposition methods, such as (hyper)tree decompositions

    The power of Sherali-Adams relaxations for general-valued CSPs

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    We give a precise algebraic characterisation of the power of Sherali-Adams relaxations for solvability of valued constraint satisfaction problems to optimality. The condition is that of bounded width which has already been shown to capture the power of local consistency methods for decision CSPs and the power of semidefinite programming for robust approximation of CSPs. Our characterisation has several algorithmic and complexity consequences. On the algorithmic side, we show that several novel and many known valued constraint languages are tractable via the third level of the Sherali-Adams relaxation. For the known languages, this is a significantly simpler algorithm than the previously obtained ones. On the complexity side, we obtain a dichotomy theorem for valued constraint languages that can express an injective unary function. This implies a simple proof of the dichotomy theorem for conservative valued constraint languages established by Kolmogorov and Zivny [JACM'13], and also a dichotomy theorem for the exact solvability of Minimum-Solution problems. These are generalisations of Minimum-Ones problems to arbitrary finite domains. Our result improves on several previous classifications by Khanna et al. [SICOMP'00], Jonsson et al. [SICOMP'08], and Uppman [ICALP'13].Comment: Full version of an ICALP'15 paper (arXiv:1502.05301
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