Dynamic Concentration in Some Discrete Random Processes

Abstract

This thesis is organized into three parts. In the first part., we analyze the random greedy algorithm for sum-free sets. A set S ⊂ [special characters omitted]_2n is sum-free if it has no solution to the equa, tion a + b = c. The random greedy algorithm starts with S := 0, and iteratively inserts elements of [special characters omitted]_2n, where each element inserted is chosen uniformly at random from those that would not. spoil the sum-free property. The algorithm terminates with a maximal sum-free set.. We show that with high probability, the algorithm produces a set of size at least [special characters omitted] n^1/2log^1/2 n. We conjecture that, this size is the correct order of magnitude (i.e. that, a matching upper bound holds, up to a constant factor).

In the second part, we analyze the random greedy algorithm for matchings in regular hypergraphs. Let r be a fixed constant and let H be an r-uniform, D-regular hypergraph on N vertices. Assume further that D → ∞ as N → ∞ and that co-degrees of pairs of vertices in H are at most L where L = o(D/ log⁵ N). We consider the random greedy algorithm for forming a matching in H. The random greedy algorithm iteratively builds a matching by choosing edges uniformly at random to be in the matching and deleting all edges that share at least one vertex with the chosen edge before moving on to the next choice. This process terminates when there are no edges remaining in the graph. We show that with high probability the proportion of vertices of H that are not saturated by the final matching is at a most (L/D) [special characters omitted]. This point is a natural barrier in the analysis of the random greedy hypergraph matching process.

In the third part, we analyze a greedy algorithm for finding large 2-matchings in 3-regular graphs. A 2-matching of a graph G is a spanning subgraph with maximum degree two. The size of a 2-matching U is the number of edges in U and this is at least n – κ( U) where n is the number of vertices of G and κ denotes the number of components. In this paper, we analyze the performance of a, greedy algorithm 2_{GREEDY f}or finding a large 2-matching on a random 3-regular graph. We prove that with high probability, the algorithm outputs a 2-matching U with κ( U) = [special characters omitted] (n^1/5).