On the power of choices in random graph processes

Abstract

The notion of the ‘power of two choices’ essentially dates back to a seminal 1994 paper by Azar, Broder, Karlin, and Upfal. Roughly speaking, their result states that if one allocates a large number of jobs to a large number of servers by assigning each job to the currently less busy of two randomly chosen servers, one observes a dramatic improvement in load balancing over a completely random assignment. This breakthrough result marked the beginning of the development of the power of two choices as a powerful new paradigm in computer science, with applications to load balancing, hashing, distributed computing, network routing, and other areas. In this thesis we are concerned with the power of choices in random graph processes. The systematic study of random graphs was initiated by Erdős and Renyi in the 60’s, and has developed into a very active research field since then. Many questions in this area involve the following random graph process: starting with the empty graph on n vertices, at every step a new edge is drawn uniformly at random from all non-edges and inserted into the current graph. Often one asks about the ‘typical’ number of edges that appear until the evolving graph satisfies some monotone property. This (asymptotically) typical number of edges is formalized in the notion of the threshold function N0(n) for a given property. In the last decades, threshold results have been proved for many natural graph properties. Several ways of introducing the power of choices into the Erdős-Renyi random graph process were proposed and studied. The most prominent process of this type is the Achlioptas process, where at each step a fixed number r ≥ 2 of random edges is offered, and one has to select exactly one of these for inclusion in the evolving graph. Bohman and Frieze showed in 2001 that in the Achlioptas process with r = 2 choices, one can significantly delay the phase transition of the evolving graph, i.e. the point where its component structure changes abruptly. Since then, the study of power-of-choice random graph processes has evolved into an active area of research in which one observes a rich interplay of combinatorial and probabilistic ideas. In this thesis we study the appearance of small subgraphs in power-of-choice random graph processes. That is, we investigate for how long one can avoid copies of some given fixed graph by exploiting the power of choices offered in a given process. Our first main result gives the general threshold function N0(F, r, n) for the problem of avoiding copies of some fixed graph F in the Achlioptas process with r choices for as long as possible. In other words, we determine the exact order of magnitude of the number of steps for which a copy of F can be avoided with high probability. This problem was studied previously by Krivelevich, Loh and Sudakov, who derived such threshold functions for some special graphs F , in particular for cliques and cycles. The general threshold function N0(F, r, n) presented in this thesis disproves a conjecture made in their work. Our second main result concerns the same problem in an offline setting, i.e. under the assumption that the entire random instance is revealed before any selection decisions have to be made. This is of interest because the offline setting provides an upper bound on what can be achieved by