QCSP Monsters and the Demise of the Chen Conjecture

TLDR: A surprising classification for the computational complexity of the Quantified Constraint Satisfaction Problem over a constraint language Γ, QCSP, where Γ is a finite language over 3 elements which contains all constants refutes the hitherto widely-believed Chen Conjecture.

Abstract: We give a surprising classification for the computational complexity of the Quantified Constraint Satisfaction Problem over a constraint language Γ, QCSP(Γ), where Γ is a finite language over three elements that contains all constants. In particular, such problems are 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 Γ such that QCSP(Γ) is DP-complete (from Boolean Hierarchy), and on a 10-element domain there exists a constraint language giving the complexity class ΘP2. Meanwhile, we prove the Chen Conjecture for finite conservative languages Γ. If the polymorphism clone of such Γ has the polynomially generated powers property, then QCSP(Γ) is in NP. Otherwise, the polymorphism clone of Γ has the exponentially generated powers property and QCSP(Γ) is PSpace-complete.1