Rapid Mixing of k-Class Biased Permutations

Abstract

In this paper, we study a biased version of the nearest-neighbor transposition Markov chain on the set of permutations where neighboring elements i and j are placed in order (i, j) with probability p i,j . Our goal is to identify the class of parameter sets P = {p i,j } for which this Markov chain is rapidly mixing. Specifically, we consider the open conjecture of Jim Fill that all monotone, positively biased distributions are rapidly mixing.We resolve Fill's conjecture in the affirmative for distributions arising from k-class particle processes, where the elements are divided into k classes and the probability of exchanging neighboring elements depends on the particular classes the elements are in. We further require that k is a constant, and all probabilities between elements in different classes are bounded away from 1/2. These particle processes arise in the context of self-organizing lists and our result also applies beyond permutations to the setting where all particles in a class are indistinguishable. Additionally we show that a broader class of distributions based on trees is also rapidly mixing, which generalizes a class analyzed by Bhakta et. al. (SODA '13). Our work generalizes recent work by Haddadan and Winkler (STACS '17) studying 3-class particle processes.Our proof involves analyzing a generalized biased exclusion process, which is a nearestneighbor transposition chain applied to a 2-particle system. Biased exclusion processes are of independent interest, with applications in self-assembly. We generalize the results of Greenberg et al. (SODA '09) and Benjamini et. al (Trans. AMS '05) on biased exclusion processes to allow the probability of swapping neighboring elements to depend on the entire system, as long as the minimum bias is bounded away from 1.The fundamental problem of generating a random permutation has a long history in computer science, beginning as early as 1969 [13]. One way to generate a random permutation is to use the nearest-neighbor Markov chain M nn which repeatedly swaps the elements in a random pair of adjacent positions. The chain M nn was among the first considered in the study of the computational efficiency of Markov chains for sampling [1,6,4] and has subsequently been studied extensively. After a series of papers, the mixing time of M nn (Θ(n 3 log n) [25]) is now well-understood when sampling from the uniform distribution on the permutation group S n .The nearest-neighbor chain M nn can also be used to sample from more general probability distributions by allowing non-uniform swap probabilities. Suppose we have a set of parameters P = {p i,j } and that M nn puts neighboring elements i and j in order (i, j) with probability p i,j , where p j,i = 1 − p i,j . Despite the simplicity of this natural extension, much less is known about the mixing time of M nn in the non-uniform case. In this paper we look at the question for which parameter sets P is M nn rapidly (polynomially) mixing? We say P is positively

Iterated Decomposition of Biased Permutations Via New Bounds on the Spectral Gap of Markov Chains

“…Here we assume there are at least C q log n particles of each type (where C q is a constant depending on the minimum bias q). We show that at each application of the iterated decomposition given in [22], the chains are 1/n-orthogonal. Thus, we can apply Theorem 3 iteratively to get a bound of Ω(n −2 ) on the spectral gap of the k-particle process M p .…”

“…The Markov chain M n has been widely studied [1,2,5,7,30]; see, e.g. [22] for a review. We say P is positively biased if p i,j ≥ 1/2 for all i < j.…”

“…Bhakta et al [2] identified certain classes of P for which M n is actually a product of independent Markov chains. Subsequently, two papers analyzed biased k-classes [14,22], where there are k classes of particles and particles from class i and class j interact with the same probability p i,j . When k = n, this is the same as the original permutation problem.…”

“…They considered bounded k-classes, where p i,j /p j,i ≤ q for all i < j for some constant q < 1. In [14], Haddadan and Winkler showed a polynomial time bound on the mixing time when k = 3 and Miracle and Streib [22] generalized this to all constant k. They proved a bound of Ω(n −2(k−1) ) on the spectral gap of M p , which after applying the comparison technique [24] and relating the gap to the mixing time, leads to a bound of O(n 2k+6 ln k) on the mixing time of the permutation process.…”

“…Here we consider the case where the probabilities P form a weakly monotone bounded k-class (introduced by [22], see Section B.1 for more details). In a bounded k-class, the set [n] can be partitioned into k particle classes C 1 , .…”