On the Chen Conjecture regarding the complexity of QCSPs

TL;DR

This paper proves a full complexity classification for the Quantified Constraint Satisfaction Problem (QCSP) based on algebraic properties, confirming the Chen Conjecture that links algebraic conditions to NP membership or co-NP hardness.

Contribution

It combines existing results to establish a complete dichotomy for QCSP complexity based on the polynomially and exponentially generated powers properties.

Findings

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

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

The paper confirms the Chen Conjecture for the complexity classification of QCSPs.

Abstract

Let A be an idempotent algebra on a finite domain. We combine results of Chen 2008 and Zhuk 2015 to argue that if Inv(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 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.