Graphs are basic combinatorial objects that are widely used in computer science, engineering, and beyond. Many problems, both theoretical and practical, can be abstracted through graphs, and graphs are also used to represent data. Several basic optimization problems on graphs naturally arise in many different contexts, and it is important to build and expand a toolkit of algorithmic ideas and techniques for addressing such problems. This project will focus on two broad classes of graph optimization problems: graph routing and graph sparsification. Graph routing problems are used as abstractions in many applications, for example, when designing VLSI chips, in multi-robot motion planning, and routing traffic across optical networks. Graph sparsification can be a useful tool in analyzing large and complex networks, including social ones, and in designing faster algorithms for a variety of graph problems. Graph routing and sparsification problems were studied in several different areas, such as approximation algorithms, fixed parameter tractability, and graph theory. In addition to designing better algorithms for such problems, another goal of the project is to increase the flow of ideas and collaboration between these different communities, by studying problems lying in the intersection of these areas. The project will involve graduate and possibly undergraduate students from TTIC and University of Chicago, as well as students from other institutions who participate in TTIC's summer internship program. The PI will also participate in activities aimed at increasing the participation of women in theoretical computer science.
Typically in a graph routing problem one is given a graph G and a collection of pairs of vertices, called demand pairs, that need to be connected to each other. The goal is to connect as many of the pairs as possible via paths, usually subject to some restrictions on the maximum load on graph vertices or edges. These are fundamental problems that have been studied extensively by both theoretical computer science and graph theory communities. Unfortunately, there are still wide gaps in our understanding of some of the most basic problems in this area, and this project aims to make progress on them. In graph sparsification problems, one is given a graph G and a small set T of its vertices called terminals. The goal is to represent the graph G by a much smaller graph H (called a sparsifier), that approximately preserves the properties of G with respect to T. The specific types of properties one would like to preserve give rise to several types of sparsifiers (for example, we may want to preserve cuts, flows or integral routings between the terminals). Problems in this area naturally connect to approximation algorithms (where sparsifiers can be used to obtain improved approximation factors to various problems), fixed-parameter tractability (where they give a black-box way to design faster algortihms), and to graph theory (many sparsification problems can be cast in graph theoretic terms and have been studied by the graph theory community). There are still many open problems in this area, and this project will attempt to improve the state of the art on several of them.
