Hierarchies of Minion Tests for PCSPs through Tensors

TL;DR

This paper introduces a unified algebraic and geometric framework for analyzing hierarchies of relaxations in PCSPs using tensor spaces, revealing new insights into their properties and connections.

Contribution

It develops a novel tensor-based approach to study hierarchies of relaxations for PCSPs, unifying various existing hierarchies under a common framework.

Findings

Characterizes the solvability of Sum-of-Squares SDP relaxation.

Identifies classes of minions with fine hierarchy features.

Unifies multiple relaxation hierarchies within the tensor framework.

Abstract

We provide a unified framework to study hierarchies of relaxations for Constraint Satisfaction Problems and their Promise variant. The idea is to split the description of a hierarchy into an algebraic part, depending on a minion capturing the "base level", and a geometric part - which we call tensorisation - inspired by multilinear algebra. We exploit the geometry of the tensor spaces arising from our construction to prove general properties of hierarchies. We identify certain classes of minions, which we call linear and conic, whose corresponding hierarchies have particularly fine features. We establish that the (combinatorial) bounded width, Sherali-Adams LP, affine IP, Sum-of-Squares SDP, and combined "LP + affine IP" hierarchies are all captured by this framework. In particular, in order to analyse the Sum-of-Squares SDP hierarchy, we also characterise the solvability of the standard SDP relaxation through a new minion.