Toward a Uniform Algorithm and Uniform Reduction for Constraint Problems

TL;DR

This paper introduces a unified framework for analyzing the power of various higher-level algorithms for constraint satisfaction problems, linking solvability to algebraic structures and proposing new SDP-like relaxations.

Contribution

It provides a minion-theoretic characterization of CSP solvability and reductions, and introduces a new hierarchy of vector relaxations equivalent to existing ones.

Findings

The vector relaxation solves the CSP of the dihedral group D4.

The p-th level of the Z_p relaxation solves linear equations modulo p^2.

Solvability depends only on the polymorphism minion of the template.

Abstract

We develop a unified framework to characterize the power of higher-level algorithms for the constraint satisfaction problem (CSP), such as $k$-consistency, the Sherali-Adams LP hierarchy, and the affine IP hierarchy. As a result, solvability of a fixed-template CSP or, more generally, a Promise CSP by a given level is shown to depend only on the polymorphism minion of the template. Similarly, we obtain a minion-theoretic description of $k$-consistency reductions between Promise CSPs. We introduce a new hierarchy of SDP-like vector relaxations with vectors over $\mathbb Z_{p}$ in which orthogonality is imposed on $k$-tuples of vectors. Surprisingly, this relaxation turns out to be equivalent to the $k$-th level of the AIP-$\mathbb{Z}_p$ relaxation. We show that it solves the CSP of the dihedral group $\mathbf{D}_4$, the smallest CSP that fools the singleton BLP+AIP algorithm. Using this vector representation, we further show that the $p$-th level of the $\mathbb{Z}_p$ relaxation solves linear equations modulo $p^2$.