By implementing algorithmic versions of Sapozhenko's graph container methods, we give new algorithms for approximating the number of independent sets in bipartite graphs. Our first algorithm applies to $d$-regular, bipartite graphs satisfying a weak expansion condition: when $d$ is constant, and the graph is a bipartite $\Omega( \log^2 d/d)$-expander, we obtain an FPTAS for the number of independent sets. Previously such a result for $d>5$ was known only for graphs satisfying the much stronger expansion conditions of random bipartite graphs. The algorithm also applies to weighted independent sets: for a $d$-regular, bipartite $\alpha$-expander, with $\alpha>0$ fixed, we give an FPTAS for the hard-core model partition function at fugacity $\lambda=\Omega(\log d / d^{1/4})$. Finally we present an algorithm that applies to all $d$-regular, bipartite graphs, runs in time $\exp\left( O\left( n \cdot \frac{ \log^3 d }{d } \right) \right)$, and outputs a $(1 + o(1))$-approximation to the number of independent sets.

中文翻译：

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通过图容器近似计算二部图中的独立集

通过实现 Sapozhenko 的图容器方法的算法版本，我们提供了用于近似二部图中独立集数量的新算法。我们的第一个算法适用于满足弱扩展条件的 $d$-regular 二部图：当 $d$ 是常数，并且图是二部 $\Omega( \log^2 d/d)$-expander 时，我们得到独立集数量的 FPTAS。以前，$d>5$ 的这种结果仅适用于满足随机二部图更强扩展条件的图。该算法也适用于加权独立集：对于 $d$-regular、二部 $\alpha$-expander，$\alpha>0$ 固定，我们给出了在逸度 $\ 的硬核模型分区函数的 FPTAS lambda=\Omega(\log d / d^{1/4})$。最后，我们提出了一个适用于所有 $d$-regular 的算法，