## Dartmouth College Ph.D Dissertations

5-1-2016

Thesis (Ph.D.)

#### Department or Program

Department of Computer Science

Peter Winkler

#### Abstract

Partially ordered sets and permutations are combinatorial structures having vast applications in theoretical computer science. In this thesis, we study various computational and algorithmic problems related to these structures. The first chapter of the thesis contains discussion about randomized fully polynomial approximation schemes obtained by employing Markov chain Monte Carlo. In this chapter we study various Markov chains that we call: the gladiator chain, the interval chain, and cube shuffling. Our objective is to identify some conditions that assure rapid mixing; and we obtain partial results. The gladiator chain is a biased random walk on the set of permutations. This chain is related to self organizing lists, and various versions of it have been studied. The interval chain is a random walk on the set of points in $\mathbb{R}^n$ whose coordinates respect a partial order. Since the sample space of the interval chain is continuous, many mixing techniques for discrete chains are not applicable to it. The cube shuffle chain is a generalization of H\r{a}stad's square shuffle. The importance of this chain is that it mixes in constant number of steps. In the second chapter, we are interested in calculating expected value of real valued function $f:S\rightarrow \mathbb{R}$ on a set of combinatorial structures $S$, given a probability distribution on it. We first suggest a Markov chain Monte Carlo approach to this problem. We identify the conditions under which our proposed solution will be efficient, and present examples where it fails. Then, we study homomesy. Homomesy is a phenomenon introduced by Jim Propp and Tom Roby. We say the triple $\langle S, \tau,f\rangle$ ($\tau$ is a permutation mapping $S$ to itself) exhibits homomesy, if the average of $f$ along all $\tau$-orbits of $S$ is a constant only depending on $f$ and $S$. We study homomesy and obtain some results when $S$ is the set of ideals in a class of simply described lattices.