The limits of SDP relaxations for general-valued CSPs

TL;DR

This paper demonstrates that for certain constraint languages, Sherali-Adams LP and Lasserre SDP hierarchies cannot efficiently solve VCSPs if the support violates the bounded width condition, highlighting fundamental limits of these relaxations.

Contribution

It proves that violating the bounded width condition prevents polynomial-size SDP relaxations from solving VCSPs, extending known inapproximability results to general-valued languages.

Findings

Violating the bounded width condition blocks Sherali-Adams LP hierarchy solutions.

Lasserre SDP hierarchy also fails for such VCSPs, even at high levels.

Reductions preserve SDP solvability, strengthening the inapproximability results.

Abstract

It has been shown that for a general-valued constraint language $\Gamma$ the following statements are equivalent: (1) any instance of $\operatorname{VCSP}(\Gamma)$ can be solved to optimality using a constant level of the Sherali-Adams LP hierarchy; (2) any instance of $\operatorname{VCSP}(\Gamma)$ can be solved to optimality using the third level of the Sherali-Adams LP hierarchy; (3) the support of $\Gamma$ satisfies the "bounded width condition", i.e., it contains weak near-unanimity operations of all arities. We show that if the support of $\Gamma$ violates the bounded width condition then not only is $\operatorname{VCSP}(\Gamma)$ not solved by a constant level of the Sherali-Adams LP hierarchy but it is also not solved by $\Omega(n)$ levels of the Lasserre SDP hierarchy (also known as the sum-of-squares SDP hierarchy). For $\Gamma$ 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 $\Gamma$ whose support violates the bounded width condition no SDP relaxation of polynomial-size solves $\operatorname{VCSP}(\Gamma)$. 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 $\{0,\infty\}$-valued languages (CSPs), $\{0,1\}$-valued languages (Min-CSPs/Max-CSPs), and $\mathbb{Q}$-valued languages (finite-valued CSPs).