Algorithmic Problems Arising in Posets and Permutations

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\u27s 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

On the Complexity of Sampling Vertices Uniformly from a Graph

We study a number of graph exploration problems in the following natural scenario: an algorithm starts exploring an undirected graph from some seed vertex; the algorithm, for an arbitrary vertex v that it is aware of, can ask an oracle to return the set of the neighbors of v. (In the case of social networks, a call to this oracle corresponds to downloading the profile page of user v.) The goal of the algorithm is to either learn something (e.g., average degree) about the graph, or to return some random function of the graph (e.g., a uniform-at-random vertex), while accessing/downloading as few vertices of the graph as possible. Motivated by practical applications, we study the complexities of a variety of problems in terms of the graph\u27s mixing time t_{mix} and average degree d_{avg} - two measures that are believed to be quite small in real-world social networks, and that have often been used in the applied literature to bound the performance of online exploration algorithms. Our main result is that the algorithm has to access Omega (t_{mix} d_{avg} epsilon^{-2} ln delta^{-1}) vertices to obtain, with probability at least 1-delta, an epsilon additive approximation of the average of a bounded function on the vertices of a graph - this lower bound matches the performance of an algorithm that was proposed in the literature. We also give tight bounds for the problem of returning a close-to-uniform-at-random vertex from the graph. Finally, we give lower bounds for the problems of estimating the average degree of the graph, and the number of vertices of the graph

Mixing of Permutations by Biased Transposition

Markov chains defined on the set of permutations of n elements have been studied widely by mathematicians and theoretical computer scientists. We consider chains in which a position i<n is chosen uniformly at random, and then sigma(i) and sigma(i+1) are swapped with probability depending on sigma(i) and sigma(i+1). Our objective is to identify some conditions that assure rapid mixing. One case of particular interest is what we call the "gladiator chain," in which each number g is assigned a "strength" s_g and when g and g\u27 are swapped, g comes out on top with probability s_g / ( s_g + s_g\u27 ). The stationary probability of this chain is the same as that of the slow-mixing "move ahead one" chain for self-organizing lists, but an open conjecture of Jim Fill\u27s implies that all gladiator chains mix rapidly. Here we obtain some positive partial results by considering cases where the gladiators fall into only a few strength classes

RePBubLik: Reducing the Polarized Bubble Radius with Link Insertions

The topology of the hyperlink graph among pages expressing different opinions may influence the exposure of readers to diverse content. Structural bias may trap a reader in a polarized bubble with no access to other opinions. We model readers' behavior as random walks. A node is in a polarized bubble if the expected length of a random walk from it to a page of different opinion is large. The structural bias of a graph is the sum of the radii of highly-polarized bubbles. We study the problem of decreasing the structural bias through edge insertions. Healing all nodes with high polarized bubble radius is hard to approximate within a logarithmic factor, so we focus on finding the best $k$ edges to insert to maximally reduce the structural bias. We present RePBubLik, an algorithm that leverages a variant of the random walk closeness centrality to select the edges to insert. RePBubLik obtains, under mild conditions, a constant-factor approximation. It reduces the structural bias faster than existing edge-recommendation methods, including some designed to reduce the polarization of a graph

The expected jaggedness of order ideals

The jaggedness of an order ideal in a poset is the number of maximal elements in plus the number of minimal elements of not in . A probability distribution on the set of order ideals of is toggle-symmetric if for every , the probability that is maximal in equals the probability that is minimal not in . In this paper, we prove a formula for the expected jaggedness of an order ideal of under any toggle-symmetric probability distribution when is the poset of boxes in a skew Young diagram. Our result extends the main combinatorial theorem of Chan-L\uf3pez-Pflueger-Teixidor [Trans. Amer. Math. Soc., forthcoming. 2015, arXiv:1506.00516], who used an expected jaggedness computation as a key ingredient to prove an algebro-geometric formula; and it has applications to homomesies, in the sense of Propp-Roby, of the antichain cardinality statistic for order ideals in partially ordered sets