超图配分函数的复杂性二分法
A complexity dichotomy for hypergraph partition functions
摘要：
研究了从$r$ -一致超图$G$到对称$r$ -元关系$H$的同态计数的复杂性.我们给出了一个二分法定理$r>2$，显示为$H$这个问题是在FP和为$H$它是#P-完全的。这推广了Dyer和Greenhill（2000）关于$r=2$情形的定理，它对应于计数图同态。我们的二分法定理扩展到关系$H$是加权的情况，目标是计算\emph{partition function}，这是同态的权重之和。这个问题的动机是统计物理学，它产生于计算粒子模型的配分函数，其中$r$位点的某些组合对称地相互作用。在加权情况下，我们的二分法定理推广了Bulatov和Grohe（2005）关于图的结果，其中$r=2$。当$r=2$时，二分法的多项式时间情况简单地对应于秩1权重。令人惊讶的是，对于所有的$r>2$，二分法的多项式时间情况具有更多的结构。事实证明，权重必须叠加在由阿贝尔群上的方程的解定义的组合结构上。我们的结果也给出了一个二分法密切相关的约束满足问题。
英文摘要：
We consider the complexity of counting homomorphisms from an $r$-uniform hypergraph $G$ to a symmetric $r$-ary relation $H$. We give a dichotomy theorem for $r>2$, showing for which $H$ this problem is in FP and for which $H$ it is #P-complete. This generalises a theorem of Dyer and Greenhill (2000) for the case $r=2$, which corresponds to counting graph homomorphisms. Our dichotomy theorem extends to the case in which the relation $H$ is weighted, and the goal is to compute the \emph{partition function}, which is the sum of weights of the homomorphisms. This problem is motivated by statistical physics, where it arises as computing the partition function for particle models in which certain combinations of $r$ sites interact symmetrically. In the weighted case, our dichotomy theorem generalises a result of Bulatov and Grohe (2005) for graphs, where $r=2$. When $r=2$, the polynomial time cases of the dichotomy correspond simply to rank-1 weights. Surprisingly, for all $r>2$ the polynomial time cases of the dichotomy have rather more structure. It turns out that the weights must be superimposed on a combinatorial structure defined by solutions of an equation over an Abelian group. Our result also gives a dichotomy for a closely related constraint satisfaction problem.
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