随机块模型中最小顶点覆盖问题的统计力学
Statistical mechanics of the minimum vertex cover problem in stochastic block models
摘要：
最小顶点覆盖（Min-VC）问题是一个著名的NP-hard问题。早期的研究表明，在平均度为$c$的Erdös-Rényi随机图上定义的问题在搜索临界点$c = e = 2.718 \ldots$以上的Min-VC集时表现出计算困难.在这里，我们解决这个困难是如何影响图的介观结构。为此，我们评估了随机块模型的困难的临界条件。我们进行了详细的检查的具体情况下，两个相同大小的社区，其特征在于在和出度，这是由$c_{\rm in}$和$c_{\rm out}$，分别表示。我们的分析的基础上的空腔方法表明，解决方案的搜索变得困难时，$c_{\rm in }+c_{\rm out} > e$，但变得容易再次当$c_{\text{out}}$是充分大于$c_{\mathrm{in}}$在区域$c_{\rm out}>e$。基于各种搜索算法的实验支持理论预测。
英文摘要：
The minimum vertex cover (Min-VC) problem is a well-known NP-hard problem. Earlier studies illustrate that the problem defined over the Erdös-Rényi random graph with a mean degree $c$ exhibits computational difficulty in searching the Min-VC set above a critical point $c = e = 2.718 \ldots$. Here, we address how this difficulty is influenced by the mesoscopic structures of graphs. For this, we evaluate the critical condition of difficulty for the stochastic block model. We perform a detailed examination of the specific cases of two equal-size communities characterized by in- and out- degrees, which are denoted by $c_{\rm in}$ and $c_{\rm out}$, respectively. Our analysis based on the cavity method indicates that the solution search becomes difficult when $c_{\rm in }+c_{\rm out} > e$, but becomes easy again when $c_{\text{out}}$ is sufficiently larger than $c_{\mathrm{in}}$ in the region $c_{\rm out}>e$. Experiments based on various search algorithms support the theoretical prediction.
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