The periodic structure of local consistency

TL;DR

The mixing behaviour of random walks over a graph is connected to the power of the local-consistency algorithm for the solution of the corresponding constraint satisfaction problem (CSP) and this connection is extended to arbitrary CSPs and their promise variant.

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 Erdős-Rényi 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.

Figures (3)

• Figure : (a) The adjacency matrix $M$ of the digraph above satisfies $M^5\sim J$, so $M$ is primitive. However, $MM^\top$ is not irreducible.
• Figure : (a) The adjacency matrix $M$ of the digraph above satisfies $M^5\sim J$, so $M$ is primitive. However, $MM^\top$ is not irreducible.
• Figure : (b) The adjacency matrix $M$ of the digraph above satisfies $(MM^\top)^2\sim J$, so $MM^\top$ is primitive. However, $M$ is not irreducible.

Theorems & Definitions (57)

• Theorem 1
• Corollary 2
• Definition 3: Aperiodicity, monic case
• Definition 4: Aperiodicity, general case
• Theorem 5
• Theorem 6
• Proposition 7
• Proposition 8
• Proposition 9
• proof : Proof of Theorem \ref{['thm_sparse_implies_large_fibrosity_plus_pendency']}