QCSP怪物和陈猜想的灭亡
QCSP monsters and the demise of the Chen Conjecture
摘要：
我们给出了一个令人惊讶的分类的计算复杂性的量化约束满足问题的约束语言$Γ$，QCSP $(Γ)$，其中$Γ$是一个有限的语言在$3$元素，其中包含所有常数。特别是，这些问题要么是P，NP-完全，co-NP-完全或PSpace-完全的。我们的分类反驳了迄今为止广泛相信的陈猜想。此外，我们还表明，在4元素域上已经存在一种约束语言$Γ$，使得QCSP $(Γ)$是DP完全的（来自布尔层次结构），并且在10元素域上存在一种约束语言，给出复杂性类$Θ_{2}^{P}$。同时，我们证明了有限保守语言$Γ$的Chen猜想。如果$Γ$的多态克隆具有多项式生成幂（PGP）属性，则QCSP $(Γ)$在NP中。另外，$Γ$的多态克隆具有指数生成幂（EGP）性质，QCSP $(Γ)$是PSpace完备的。
英文摘要：
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 $3$ 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 $Γ$ 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 $Θ_{2}^{P}$. 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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