Statistical Mechanics of maximal independent sets

Abstract

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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Cited by in corpus (13)

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• The statistical mechanics of random set packing and a generalization of the Karp-Sipser algorithm
• Statistical physics approach to graphical games: local and global interactions
• Fixation and escape times in stochastic game learning
• Generalized minimum dominating set and application in automatic text summarization
• Counting and Hardness-of-Finding Fixed Points in Cellular Automata on Random Graphs
• Statics and dynamics of selfish interactions in distributed service systems
• Effect of Constraint Relaxation on the Minimum Vertex Cover Problem in Random Graphs
• Coordination problems on networks revisited: statics and dynamics
• Constructing Concrete Hard Instances of the Maximum Independent Set Problem
• Approaching the ground states of the random maximum two-satisfiability problem by a greedy single-spin flipping process
• Extended Yard Sale model of wealth distribution on Erdős-Rényi random networks
• Solving Graph-based Public Good Games with Tree Search and Imitation Learning