d-正则图上双自旋模型计数的计算困难性
The computational hardness of counting in two-spin models on d-regular graphs
摘要：
双自旋系统类包含了几个重要的模型，包括随机独立集和统计物理的伊辛模型。我们证明了对于硬核（独立集）模型和反铁磁Ising模型，当模型在d-正则树上具有非唯一性时，在d-正则图上近似模型的分拆函数或近似采样都是NP-困难的.再加上Jerrum-Sinclair，Weitz和Sinclair-Srivastava-Thurley给出的除唯一性阈值外的所有其他双自旋系统的FPRAS的结果，这给出了有界度图上双自旋系统的计算复杂性的几乎完整的分类。证明了在二分局部树状图上任意两个自旋系统的归一化对数分拆函数收敛于一个极限“自由能密度”，这与统计物理的Bethe（非严格）预言相一致.我们使用这个结果来描述局部树状二部扩展图上的两个自旋系统的局部结构，然后成为随机约简的基本工具，以近似MAX-CUT。我们的方法是新颖的，因为它没有使用二阶矩方法在以前的作品中对这些问题。
英文摘要：
The class of two-spin systems contains several important models, including random independent sets and the Ising model of statistical physics. We show that for both the hard-core (independent set) model and the anti-ferromagnetic Ising model with arbitrary external field, it is NP-hard to approximate the partition function or approximately sample from the model on d-regular graphs when the model has non-uniqueness on the d-regular tree. Together with results of Jerrum--Sinclair, Weitz, and Sinclair--Srivastava--Thurley giving FPRAS's for all other two-spin systems except at the uniqueness threshold, this gives an almost complete classification of the computational complexity of two-spin systems on bounded-degree graphs. Our proof establishes that the normalized log-partition function of any two-spin system on bipartite locally tree-like graphs converges to a limiting "free energy density" which coincides with the (non-rigorous) Bethe prediction of statistical physics. We use this result to characterize the local structure of two-spin systems on locally tree-like bipartite expander graphs, which then become the basic gadgets in a randomized reduction to approximate MAX-CUT. Our approach is novel in that it makes no use of the second moment method employed in previous works on these questions.
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