The complexity of quantified constraints using the algebraic formulation

TL;DR

This paper establishes a full complexity dichotomy for the quantified constraint satisfaction problem (QCSP) over finite domains using algebraic properties, linking PGP and EGP to NP and co-NP-hardness, and explores the special case of three-element domains.

Contribution

It proves a complete complexity classification for QCSP based on algebraic properties and clarifies the relationship between switchability and collapsibility in three-element domains.

Findings

QCSP(Inv(A)) is in NP if A satisfies PGP.

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

For three-element domains, switchability implies collapsibility, simplifying tractability analysis.

Abstract

Let A be an idempotent algebra on a finite domain. We combine results of Chen, Zhuk and Carvalho et al. to argue that if A satisfies the polynomially generated powers property (PGP), then QCSP(Inv(A)) is in NP. We then use the result of Zhuk to prove a converse, that if QCSP(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 the moral correctness of what we term the Chen Conjecture. We examine in closer detail the situation for domains of size three. Over any finite domain, the only type of PGP that can occur is switchability. Switchability was introduced by Chen as a generalisation of the already-known Collapsibility. For three-element domain algebras A that are Switchable, we prove that for every finite subset Delta of Inv(A), Pol(Delta) is Collapsible. The significance of this is that, for QCSP on finite structures (over three-element domain), all QCSP tractability explained by Switchability is already explained by Collapsibility. Finally, we present a three-element domain complexity classification vignette, using known as well as derived results.