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 width condition then not only is VCSP(Γ) not solved by a constant level of the Sherali-Adams LP hierarchy but it requires linear 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).The dichotomy conjecture of Feder and Vardi has been verified in several important special cases by Schaefer [47], Hell and Nešetřil [27], Bulatov [8,11], and Barto, Kozik, and Niven [6] mostly using the so-called algebraic approach [10,4]. Remarkably, the dichotomy conjecture has recently been solved independently by Bulatov [9] and Zhuk [58], respectively.Using concepts from the extensions of the algebraic approach to optimisation problems [17], the exact solvability of purely optimisation CSPs, known as finite-valued CSPs, has been established by the authors [50] (these include Min/Max-CSPs as a special case). Putting together decision and optimisation problems in one framework, the exact complexity of socalled general-valued CSPs has been established, modulo the (now proved) classification of decision CSPs, by the works of Kozik and Ochremiak [35] and Kolmogorov, Krokhin, and Rolínek [31]. A result that proved useful when classifying both finite-valued and generalvalued CSPs is an algebraic characterisation of the power of the basic linear programming relaxation for decision CSPs [36] and general-valued CSPs [32].
ApproximationConvex relaxations, such as linear programming (LP) and semidefinite programming (SDP), have long been powerful tools for designing efficient exact and approximation algorithms [55,56]. In particular, for many combinatorial problems, the introduction of semidefinite programming relaxations allowed for a new structural and computational perspective [23, 30...