The periodic structure of local consistency

TL;DR

This paper links the mixing properties of random walks on graphs to the effectiveness of local consistency algorithms in solving CSPs, establishing optimal lower bounds and classifying their power for certain problems.

Contribution

It extends the connection between random walk mixing and local consistency to arbitrary CSPs, proving linear lower bounds for aperiodic promise CSPs and classifying their computational power.

Findings

Established linear lower bounds for local consistency on aperiodic promise CSPs.

Classified the power of local consistency for approximate graph homomorphism problems.

Connected graph mixing behavior to the complexity of solving related CSPs.

Abstract

We connect the mixing behaviour of random walks over a graph to the power of the local-consistency algorithm for the solution of the corresponding constraint satisfaction problem (CSP). We extend this connection to arbitrary CSPs and their promise variant. In this way, we establish a linear-level (and, thus, optimal) lower bound against the local-consistency algorithm applied to the class of aperiodic promise CSPs. The proof is based on a combination of the probabilistic method for random Erd\H{o}s-R\'enyi hypergraphs and a structural result on the number of fibers (i.e., long chains of hyperedges) in sparse hypergraphs of large girth. As a corollary, we completely classify the power of local consistency for the approximate graph homomorphism problem by establishing that, in the nontrivial cases, the problem has linear width.