The complexity of quantified constraints: collapsibility, switchability and the algebraic formulation
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 complexitytheoretic 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 coNPhard. 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 alreadyknown collapsibility. For threeelement domain algebras A that are switchable and omit a Gset, 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 threeelement domain), all QCSP tractability (in P) explained by switchability is already explained by collapsibility.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 DOI:
 10.48550/arXiv.2106.13154
 arXiv:
 arXiv:2106.13154
 Bibcode:
 2021arXiv210613154C
 Keywords:

 Computer Science  Computational Complexity
 EPrint:
 arXiv admin note: substantial text overlap with arXiv:1701.04086, arXiv:1501.04558, arXiv:1510.06298