Approximately Sampling Elements with Fixed Rank in Graded Posets

Graded posets frequently arise throughout combinatorics, where it is natural to try to count the number of elements of a fixed rank. These counting problems are often $\#\textbf{P}$-complete, so we consider approximation algorithms for counting and uniform sampling. We show that for certain classes of posets, biased Markov chains that walk along edges of their Hasse diagrams allow us to approximately generate samples with any fixed rank in expected polynomial time. Our arguments do not rely on the typical proofs of log-concavity, which are used to construct a stationary distribution with a specific mode in order to give a lower bound on the probability of outputting an element of the desired rank. Instead, we infer this directly from bounds on the mixing time of the chains through a method we call $\textit{balanced bias}$. A noteworthy application of our method is sampling restricted classes of integer partitions of $n$. We give the first provably efficient Markov chain algorithm to uniformly sample integer partitions of $n$ from general restricted classes. Several observations allow us to improve the efficiency of this chain to require $O(n^{1/2}\log(n))$ space, and for unrestricted integer partitions, expected $O(n^{9/4})$ time. Related applications include sampling permutations with a fixed number of inversions and lozenge tilings on the triangular lattice with a fixed average height.Comment: 23 pages, 12 figure

Markov chains for weighted lattice structures

Markov chains are an essential tool for sampling from large sets, and are ubiquitous across many scientific fields, including statistical physics, industrial engineering, and computer science. To be a useful tool for sampling, the number of steps needed for a Markov chain to converge approximately to the target probability distribution, also known as the mixing time, should be a small polynomial in n, the size of a state. We study problems that arise from the design and analysis of Markov chains that sample from configurations of lattice structures. Specifically, we will be interested in settings where each state is sampled with a non-uniform weight that depends on the structure of the configuration. These weighted lattice models arise naturally in many contexts, and are typically more difficult to analyze than their unweighted counterparts. Our focus will be on exploiting these weightings both to develop new efficient algorithms for sampling and to prove new mixing time bounds for existing Markov chains. First, we will present an efficient algorithm for sampling fixed rank elements from a graded poset, which includes sampling integer partitions of n as a special case. Then, we study the problem of sampling weighted perfect matchings on lattices using a natural Markov chain based on "rotations", and provide evidence towards understanding why this Markov chain has empirically been observed to converge slowly. Finally, we present and analyze a generalized version of the Schelling Segregation model, first proposed in 1971 by economist Thomas Schelling to explain possible causes of racial segregation in cities. We identify conditions under which segregation, or clustering, is likely or unlikely to occur. Our analysis techniques for all three problems are drawn from the interface of theoretical computer science with discrete mathematics and statistical physics.Ph.D

Cut-Colorings in Coloring Graphs

This paper studies the connectivity and biconnectivity of coloring graphs. For kN, the k-coloring graph of a base graph G has vertex set consisting of the proper k-colorings of G and edge set consisting of the pairs of k-colorings that differ on a single vertex. A base graph whose k-coloring graph is connected is said to be k-mixing; it is possible to transition between any two k-colorings in a k-mixing graph via a sequence of single vertex recolorings, where each intermediate step is also a proper k-coloring. A natural extension of connectedness is biconnectedness. If a base graph has a coloring graph that is not biconnected, then there exists a proper k-coloring that would disconnect the coloring graph if removed. We call such a coloring a k-cut coloring. We prove that no base graph that is 3-mixing can have a 3-cut coloring, but for any k4 there exists a base graph that is k-mixing and has a k-cut coloring

Mixing Times of Markov Chains for Self-Organizing Lists and Biased Permutations

We study the mixing time of a Markov chain Mnn on permutations that performs nearest neighbor transpositions in the non-uniform setting, a problem arising in the context of self-organizing lists. We are given “positively biased ” probabilities {pi,j ≥ 1/2} for all i < j and let pj,i = 1 − pi,j. In each step, the chain Mnn chooses two adjacent elements k, and ℓ and exchanges their positions with probability pℓ,k. Here we define two general classes and give the first proofs that the chain is rapidly mixing for both. In the first case we are given constants r1,... rn−1 with 1/2 ≤ ri ≤ 1 for all i and we set pi,j = ri for all i < j. In the second we are given a binary tree with n leaves labeled 1,... n and constants q1,... qn−1 associated with all of the internal vertices, and we let pi,j = qi∧j for all i < j. Our bounds on the mixing time of Mnn rely on bijections between permutations, inversion tables and asymmetric simple exclusion processes (ASEPs) that allow us to express moves of the chain in the context of these other combinatorial families. We also demonstrate that the chain is not always rapidly mixing by constructing an example requiring exponential time to converge to equilibrium. This proof relies on a reduction to biased lattice paths in Z²