On the hardness of sampling independent sets beyond the tree threshold

• 成果类型: Article
• 署名作者: Mossel, Elchanan; Weitz, Dror; Wormald, Nicholas
• 署名单位: Rutgers University System; Rutgers University New Brunswick; University of Waterloo
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-007-0131-9
• 发表日期: 2009
• 页码: 401-439

摘要

We consider local Markov chain Monte-Carlo algorithms for sampling from the weighted distribution of independent sets with activity lambda, where the weight of an independent set I is lambda(|I|). A recent result has established that Gibbs sampling is rapidly mixing in sampling the distribution for graphs of maximum degree d and lambda < lambda (c) (d), where lambda (c) (d) is the critical activity for uniqueness of the Gibbs measure (i.e., for decay of correlations with distance in the weighted distribution over independent sets) on the d-regular infinite tree. We show that for d a parts per thousand yen 3, lambda just above lambda (c) (d) with high probability over d-regular bipartite graphs, any local Markov chain Monte-Carlo algorithm takes exponential time before getting close to the stationary distribution. Our results provide a rigorous justification for replica method heuristics. These heuristics were invented in theoretical physics and are used in order to derive predictions on Gibbs measures on random graphs in terms of Gibbs measures on trees. A major theoretical challenge in recent years is to provide rigorous proofs for the correctness of such predictions. Our results establish such rigorous proofs for the case of hard-core model on bipartite graphs. We conjecture that lambda (c) is in fact the exact threshold for this computational problem, i.e., that for lambda > lambda (c) it is NP-hard to approximate the above weighted sum over independent sets to within a factor polynomial in the size of the graph.