极大独立集的统计力学
Statistical Mechanics of maximal independent sets
摘要：
最大独立集的图论概念出现在计算机科学和博弈论中的几个实际问题中。最大独立集是由满足某些填充和覆盖约束的被占用节点的集合定义的。众所周知，寻找最小和最大密度极大独立集是一个困难的优化问题。本文利用统计物理中的腔方法和蒙特卡罗模拟研究了随机图上相应的约束满足问题。我们得到的副本对称和一步复本对称性破缺框架内的最大独立集的熵，揭示的度量结构的解决方案，并提出了一类可能的算法。这是特别相关的应用研究战略互动的社会和经济网络，其中最大的独立集对应于纯纳什均衡的图形游戏的公共物品分配。
英文摘要：
The graph theoretic concept of maximal independent set arises in several practical problems in computer science as well as in game theory. A maximal independent set is defined by the set of occupied nodes that satisfy some packing and covering constraints. It is known that finding minimum and maximum-density maximal independent sets are hard optimization problems. In this paper, we use cavity method of statistical physics and Monte Carlo simulations to study the corresponding constraint satisfaction problem on random graphs. We obtain the entropy of maximal independent sets within the replica symmetric and one-step replica symmetry breaking frameworks, shedding light on the metric structure of the landscape of solutions and suggesting a class of possible algorithms. This is of particular relevance for the application to the study of strategic interactions in social and economic networks, where maximal independent sets correspond to pure Nash equilibria of a graphical game of public goods allocation.
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