The complexity of quantified constraints: collapsibility, switchability and the algebraic formulation

TL;DR

This paper explores the algebraic properties influencing the complexity of quantified constraints, establishing a dichotomy that clarifies when QCSP problems are tractable or hard, based on properties like PGP, EGP, switchability, and collapsibility.

Contribution

It proves a full complexity dichotomy for QCSP based on algebraic properties, confirming the Revised Chen Conjecture and relating switchability to collapsibility for three-element domains.

Findings

QCSP(B) is in NP if A satisfies PGP and B is invariant under A

QCSP(Inv(A)) is co-NP-hard if Inv(A) satisfies EGP

Switchability implies collapsibility for certain three-element domain algebras

Abstract

Let A be an idempotent algebra on a finite domain. By mediating between results of Chen and Zhuk, we argue that if A satisfies the polynomially generated powers property (PGP) and B is a constraint language invariant under A (that is, in Inv(A)), then QCSP(B) is in NP. In doing this we study the special forms of PGP, switchability and collapsibility, in detail, both algebraically and logically, addressing various questions such as decidability on the way. We then prove a complexity-theoretic converse in the case of infinite constraint languages encoded in propositional logic, that if Inv(A) satisfies the exponentially generated powers property (EGP), then QCSP(Inv(A)) is co-NP-hard. Since Zhuk proved that only PGP and EGP are possible, we derive a full dichotomy for the QCSP, justifying what we term the Revised Chen Conjecture. This result becomes more significant now the original Chen Conjecture is known to be false. Switchability was introduced by Chen as a generalisation of the already-known collapsibility. For three-element domain algebras A that are switchable and omit a G-set, we prove that, for every finite subset D of Inv(A), Pol(D) is collapsible. The significance of this is that, for QCSP on finite structures (over a three-element domain), all QCSP tractability (in P) explained by switchability is already explained by collapsibility.