Low-level dichotomy for Quantified Constraint Satisfaction Problems

TL;DR

This paper establishes a dichotomy for the complexity of quantified constraint satisfaction problems (QCSPs) over finite core structures, showing they are either very efficiently solvable or computationally hard, based on properties of the underlying CSP.

Contribution

It proves a dichotomy for QCSP(B) over finite core structures, linking the complexity to first-order expressibility of CSP(B) and demonstrating equivalences under logspace reductions.

Findings

QCSP(B) is in ALogtime if CSP(B) is first-order expressible

QCSP(B) is L-hard otherwise

Existence of a structure C with trivially true CSP(B) and equivalent QCSPs

Abstract

Building on a result of Larose and Tesson for constraint satisfaction problems (CSP s), we uncover a dichotomy for the quantified constraint satisfaction problem QCSP(B), where B is a finite structure that is a core. Specifically, such problems are either in ALogtime or are L-hard. This involves demonstrating that if CSP(B) is first-order expressible, and B is a core, then QCSP(B) is in ALogtime. We show that the class of B such that CSP(B) is first-order expressible (indeed, trivially true) is a microcosm for all QCSPs. Specifically, for any B there exists a C such that CSP(C) is trivially true, yet QCSP(B) and QCSP(C) are equivalent under logspace reductions.