It has been shown that for a general-valued constraint language Γ the following statements are equivalent: (1) any instance of VCSP(Γ) can be solved to optimality using a constant level of the Sherali-Adams LP hierarchy; (2) any instance of VCSP(Γ) can be solved to optimality using the third level of the Sherali-Adams LP hierarchy; (3) the support of Γ satisfies the "bounded width condition", i.e., it contains weak near-unanimity operations of all arities. We show that if the support of Γ violates the bounded with condition then not only is VCSP(Γ) not solved by a constant level of the Sherali-Adams LP hierarchy but it is also not solved by Ω(n) levels of the Lasserre SDP hierarchy (also known as the sum-of-squares SDP hierarchy). For Γ corresponding to linear equations in an Abelian group, this result follows from existing work on inapproximability of Max-CSPs. By a breakthrough result of Lee, Raghavendra, and Steurer [STOC'15], our result implies that for any Γ whose support violates the bounded width condition no SDP relaxation of polynomial-size solves VCSP(Γ). We establish our result by proving that various reductions preserve exact solvability by the Lasserre SDP hierarchy (up to a constant factor in the level of the hierarchy). Our results hold for general-valued constraint languages, i.e., sets of functions on a fixed finite domain that take on rational or infinite values, and thus also hold in notable special cases of 0, ∞ -valued languages (CSPs), 0, 1 -valued languages (Min-CSPs Max-CSPs), and Q-valued languages (finite-valued CSPs).
In our main result, Theorem , the BWC is required to hold, as in @cite_53 , for the support of the fractional polymorphisms @cite_49 of the general-valued CSPs. Intuitively, fractional polymorphisms of a general-valued CSP instance @math are probability distributions over polymorphisms of @math with some desirable properties. (A formal definition is given in .) This is a natural requirement since polymorphisms do not capture the complexity of general-valued CSPs but the fractional polymorphisms do so @cite_49 @cite_25 .
@cite_53 @cite_25 @cite_49
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We give a precise algebraic characterization of the power of Sherali--Adams relaxations for solvability of valued constraint satisfaction problems (CSPs) 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 characterization has several algorithmic and complexity consequences. On the algorithmic side, we show that several novel and well-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 those previously obtained. 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 Živný [J. ACM, 60 (2013), 10], and also a ... 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 infeasible assignments. The goal is to find an assignment of labels to the variables that minimizes the sum. We study (assuming that P a#x2260; 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, a#x221E; 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 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 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. Discrete optimization problems arise in many different areas and are studied under many different names. In many such problems the quantity to be optimized can be expressed as a sum of functions of a restricted form. Here we present a unifying theory of complexity for problems of this kind. We show that the complexity of a finite-domain discrete optimization problem is determined by certain algebraic properties of the objective function, which we call weighted polymorphisms. We define a Galois connection between sets of rational-valued functions and sets of weighted polymorphisms and show how the closed sets of this Galois connection can be characterized. These results provide a new approach to studying the complexity of discrete optimization. We use this approach to identify certain maximal tractable subproblems of the general problem and hence derive a complete classification of complexity for the Boolean case.