# QCSP monsters and the demise of the Chen Conjecture

## TL;DR

This paper classifies the computational complexity of QCSP over finite languages with constants, refutes the Chen Conjecture, and identifies new complexity classes for specific domains, providing a comprehensive complexity landscape.

## Contribution

It provides a complete complexity classification for QCSP over finite languages with constants, disproves the Chen Conjecture, and introduces new complexity results for specific domain sizes.

## Key findings

- QCSP over certain languages can be P, NP-complete, co-NP-complete, or PSpace-complete.
- Existence of languages with DP-complete and IGMA complexity on small domains.
- Chen Conjecture holds for finite conservative languages with specific polymorphism properties.

## Abstract

We give a surprising classification for the computational complexity of the Quantified Constraint Satisfaction Problem over a constraint language $\Gamma$, QCSP$(\Gamma)$, where $\Gamma$ is a finite language over $3$ elements which contains all constants. In particular, such problems are either in P, NP-complete, co-NP-complete or PSpace-complete. Our classification refutes the hitherto widely-believed Chen Conjecture. Additionally, we show that already on a 4-element domain there exists a constraint language $\Gamma$ such that QCSP$(\Gamma)$ is DP-complete (from Boolean Hierarchy), and on a 10-element domain there exists a constraint language giving the complexity class $\Theta_{2}^{P}$. Meanwhile, we prove the Chen Conjecture for finite conservative languages $\Gamma$. If the polymorphism clone of $\Gamma$ has the polynomially generated powers (PGP) property then QCSP$(\Gamma)$ is in NP. Otherwise, the polymorphism clone of $\Gamma$ has the exponentially generated powers (EGP) property and QCSP$(\Gamma)$ is PSpace-complete.