超顶点覆盖问题的统计力学
Statistical Mechanics of the Hyper Vertex Cover Problem
摘要：
我们介绍和研究一个新的优化问题称为超顶点覆盖。这个问题是标准顶点覆盖到超图的一个推广：寻找一个具有最小密度的粒子配置，使得超图的每个超边至少包含一个粒子。它还可以用于重要的实际任务，例如分组测试程序，其中人们希望通过池测试来检测大组中的缺陷产品。利用基于腔方法的统计力学方法，研究了随机正则超图情形下HVC问题的相图。根据变量的值和测试程度，可能会出现不同的情况：HVC问题可以是在副本对称阶段，或在一个一步复本对称性破缺之一。在这两种情况下，我们给出了明确的结果，粒子的最小密度，和相空间的结构。因此，这些问题在某种意义上是简单的比原来的顶点覆盖问题，其中需要一个完整的复本对称性破缺阻止了精确的结果推导到目前为止。最后，我们表明，抽取程序的基础上的信念传播和调查传播算法提供了非常有效的策略，以解决大型个人的超顶点覆盖问题。
英文摘要：
We introduce and study a new optimization problem called Hyper Vertex Cover. This problem is a generalization of the standard vertex cover to hypergraphs: one seeks a configuration of particles with minimal density such that every hyperedge of the hypergraph contains at least one particle. It can also be used in important practical tasks, such as the Group Testing procedures where one wants to detect defective items in a large group by pool testing. Using a Statistical Mechanics approach based on the cavity method, we study the phase diagram of the HVC problem, in the case of random regualr hypergraphs. Depending on the values of the variables and tests degrees different situations can occur: The HVC problem can be either in a replica symmetric phase, or in a one-step replica symmetry breaking one. In these two cases, we give explicit results on the minimal density of particles, and the structure of the phase space. These problems are thus in some sense simpler than the original vertex cover problem, where the need for a full replica symmetry breaking has prevented the derivation of exact results so far. Finally, we show that decimation procedures based on the belief propagation and the survey propagation algorithms provide very efficient strategies to solve large individual instances of the hyper vertex cover problem.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
