On Markov Chains for Independent Sets

Abstract

Random independent sets in graphs arise, for example, in statistical physics, in the hardcore model of a gas. In 1997, Luby and Vigoda described a rapidly mixing Markov chain for independent sets, which we refer to as the Luby–Vigoda chain. A new rapidly mixing Markov chain for independent sets is defined in this paper. Using path coupling, we obtain a polynomial upper bound for the mixing time of the new chain for a certain range of values of the parameter λ. This range is wider than the range for which the mixing time of the Luby–Vigoda chain is known to be polynomially bounded. Moreover, the upper bound on the mixing time of the new chain is always smaller than the best known upper bound on the mixing time of the Luby–Vigoda chain for larger values of λ (unless the maximum degree of the graph is 4). An extension of the chain to independent sets in hypergraphs is described. This chain gives an efficient method for approximately counting the number of independent sets of hypergraphs with maximum degree two, or with maximum degree three and maximum edge size three. Finally, we describe a method which allows one, under certain circumstances, to deduce the rapid mixing of one Markov chain from the rapid mixing of another, with the same state space and stationary distribution. This method is applied to two Markov chains for independent sets, a simple insert/delete chain, and the new chain, to show that the insert/delete chain is rapidly mixing for a wider range of values of λ than was previously known.