GMSNP and Finite Structures

TL;DR

This paper establishes a connection between GMSNP definability of certain CSPs and the existence of minimal finite factors, leading to new insights on tractability and complexity of related problems.

Contribution

It proves that if CSP(S) is in GMSNP, then S has a minimal finite factor C, and CSP(C) reduces to CSP(S), with applications to PCSPs and graph coloring problems.

Findings

Existence of minimal finite factors for structures with GMSNP CSPs.

Polynomial-time reduction from CSP(C) to CSP(S).

GMSNP CSPs for certain graphs are NP-complete.

Abstract

Given an (infinite) relational structure $\mathbb S$, we say that a finite structure $\mathbb C$ is a minimal finite factor of $\mathbb S$ if for every finite structure $\mathbb A$ there is a homomorphism $\mathbb S\to \mathbb A$ if and only if there is a homomorphism $\mathbb{C} \to \mathbb{A}$. In this brief note we prove that if CSP($\mathbb S$) is in GMSNP, then $\mathbb S$ has a minimal finite factor $\mathbb C$, and moreover, CSP($\mathbb C$) reduces in polynomial time to CSP($\mathbb S$). We discuss two nice applications of this result. First, we see that if a finite promise constraint satisfaction problem PCSP($\mathbb A,\mathbb B$) has a tractable GMSNP sandwich, then it has a tractable finite sandwich. We also show that if $\mathbb G$ is a non-bipartite (possibly infinite) graph with finite chromatic number, and CSP($\mathbb G$) is in GMSNP, then CSP($\mathbb G$) in NP-complete, partially answering a question recently asked by Bodirsky and Guzm\'an-Pro.