QCSP monsters and the demise of the Chen Conjecture
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We give a surprising classification for the computational complexity of Quantified Constraint Satisfaction Problems, QCSP(Γ), where Γ 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 4-element domain there exists a constraint language Γ such that QCSP(Γ) is DP-complete (from Boolean Hierarchy), and on 10-element domain there exists a constraint language giving a complexity class different from all the above classes. Meanwhile, we prove the Chen Conjecture for finite conservative languages Γ. If the polymorphism clone of Γ has the polynomially generated powers (PGP) property then QCSP(Γ) is in NP. Otherwise, the polymorphism clone of Γ has the exponentially generated powers (EGP) property and QCSP(Γ) is Pspace-complete.
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