We determine the computational complexity of approximately counting and sampling independent sets of a given size in bounded-degree graphs. That is, we identify a critical density $\alpha_c(\Delta)$ and provide (i) for $\alpha < \alpha_c(\Delta)$ randomized polynomial-time algorithms for approximately sampling and counting independent sets of given size at most $\alpha n$ in $n$-vertex graphs of maximum degree $\Delta$; and (ii) a proof that unless NP=RP, no such algorithms exist for $\alpha>\alpha_c(\Delta)$. The critical density is the occupancy fraction of hard core model on the clique $K_{\Delta+1}$ at the uniqueness threshold on the infinite $\Delta$-regular tree, giving $\alpha_c(\Delta)\sim\frac{e}{1+e}\frac{1}{\Delta}$ as $\Delta\to\infty$.

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在有界图中近似计算给定大小的独立集合

我们确定在有限度图中对给定大小的独立集合进行近似计数和采样的计算复杂性。也就是说，我们确定临界密度$ \ alpha_c（\ Delta）$并为$ \ alpha <\ alpha_c（\ Delta）$随机多项式时间算法提供（i），以便近似采样和计数给定大小的独立集合最大度数$ \ Delta $的$ n $顶点图中的$ \ alpha n $; （ii）证明除非NP = RP，否则对于\\ alpha> \ alpha_c（\ Delta）$不存在这样的算法。临界密度是无穷$ \ Delta $规则树上唯一性阈值上的$ K _ {\ Delta + 1} $上硬核模型的占用率，得出$ \ alpha_c（\ Delta）\ sim \ frac {e} {1 + e} \ frac {1} {\ Delta} $作为$ \ Delta \ to \ infty $。