A universal-algebraic proof of the complexity dichotomy for Monotone Monadic SNP

TL;DR

This paper provides a new algebraic proof of the complexity dichotomy for MMSNP, a logic related to graph queries, and verifies the Bodirsky-Pinsker conjecture for these problems.

Contribution

It introduces a novel proof avoiding probabilistic reductions and confirms the Bodirsky-Pinsker conjecture for MMSNP CSPs using universal algebra.

Findings

Established a new reduction proof for MMSNP to finite-domain CSPs.

Verified the Bodirsky-Pinsker dichotomy conjecture for MMSNP.

Applied universal-algebraic methods to analyze MMSNP complexity.

Abstract

The logic MMSNP is a restricted fragment of existential second-order logic which allows to express many interesting queries in graph theory and finite model theory. The logic was introduced by Feder and Vardi who showed that every MMSNP sentence is computationally equivalent to a finite-domain constraint satisfaction problem (CSP); the involved probabilistic reductions were derandomized by Kun using explicit constructions of expander structures. We present a new proof of the reduction to finite-domain CSPs which does not rely on the results of Kun. This new proof allows us to obtain a stronger statement and to verify the more general Bodirsky-Pinsker dichotomy conjecture for CSPs in MMSNP. Our approach uses the fact that every MMSNP sentence describes a finite union of CSPs for countably infinite $\omega$-categorical structures; moreover, by a recent result of Hubi\v{c}ka and Ne\v{s}et\v{r}il, these structures can be expanded to homogeneous structures with finite relational signature and the Ramsey property. This allows us to use the universal-algebraic approach to study the computational complexity of MMSNP.