The Limits of SDP Relaxations for General-Valued CSPs

Abstract

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...

Proof Complexity Meets Algebra

“…Therefore, Theorem 5 implies the following statement, which will be crucial for obtaining lower bounds in Section 6: One more consequence of Theorem 5 concerns proofs in Frege proof system without any bounds on the depth. Corollary 3 above immediately implies that Frege is well-behaved with respect to the classical CSP reductions, that is: In the case of algebraic proof systems, if E and E ′ are any local algebraic encoding schemes over a field F for CSP(B) and CSP(B ′ ), respectively, we show that: We point out that Theorem 7 in the case of the Sherali-Adams and Sums-of-Squares proof systems and the EQ encoding scheme can be extracted from [48] and [47].…”

“…The goal of this section is to prove the following: The equivalence of 1 and 4 is known [47]. Here we provide an alternative proof.…”

“…Any improvement on this would improve the approximation status of all problems that reduce to it, and refute the UGC [34]. For the constraint satisfaction problem, in which all constraints must be satisfied, as well as for its optimisation version, the analogue question was resolved recently also by exploiting the theory of reducibility: in that arena, low-degree SOS unsatisfiability proofs exist only for problems of bounded width [47,25].…”

The Combined Basic LP and Affine IP Relaxation for Promise VCSPs on Infinite Domains

“…Convex relaxations have been also successfully applied to the study of the three extensions of CSPs. For VCSPs, characterisations of the applicability of the basic linear programming relaxation [36], constant levels of the Sherali-Adams linear programming hierarchy [46], and a polynomial-size semidefinite programming relaxation [47] have been provided for exact solvability. In the PCSP framework, the polynomial-time tractability via a specific convex relaxation has been characterised for the basic linear programming relaxation [21], affine integer programming relaxation [21], and their combination [16,17,18].…”

Optimization Over the Boolean Hypercube Via Sums of Nonnegative Circuit Polynomials

“…Other SOS degree lower bounds for Knapsack problems appeared in [13,37]. Some lower bounds on the effectiveness of SOS have been shown for CSP problems [33,61] and for the Planted Clique Problem [3,47]. For a polynomially solvable problem of scheduling unit size jobs on a single machine to minimize the number of late jobs, a degree Ω( √ n) SOS, applied on the natural, widely used, formulation of the problem, does not provide a relaxation with bounded integrality gap [38].…”