New geometric techniques for linear programming and graph partitioning

Abstract

In this thesis, we advance a collection of new geometric techniques for the analysis of combinatorial algorithms. Using these techniques, we resolve several longstanding questions in the theory of linear programming, polytope theory, spectral graph theory, and graph partitioning. The thesis consists of two main parts. In the first part, which is joint work with Daniel Spielman, we present the first randomized polynomial-time simplex algorithm for linear programming, answering a question that has been open for over fifty years. Like the other known polynomial-time algorithms for linear programming, its running time depends polynomially on the number of bits used to represent its input. To do this, we begin by reducing the input linear program to a special form in which we merely need to certify boundedness of the linear program. As boundedness does not depend upon the right-hand-side vector, we run a modified version of the shadow-vertex simplex method in which we start with a random right-hand-side vector and then modify this vector during the course of the algorithm. This allows us to avoid bounding the diameter of the original polytope.

(cont.) Our analysis rests on a geometric statement of independent interest: given a polytope ... in isotropic position, if one makes a polynomially small perturbation to b then the number of edges of the projection of the perturbed polytope onto a random 2-dimensional subspace is expected to be polynomial. In the second part of the thesis, we address two long-open questions about finding good separators in graphs of bounded genus and degree: 1. It is a classical result of Gilbert, Hutchinson, and Tarjan [25] that one can find asymptotically optimal separators on these graphs if he is given both the graph and an embedding of it onto a low genus surface. Does there exist a simple, efficient algorithm to find these separators given only the graph and not the embedding? 2. In practice, spectral partitioning heuristics work extremely well on these graphs. Is there a theoretical reason why this should be the case? We resolve these two questions by showing that a simple spectral algorithm finds separators of cut ratio O( g/n) and vertex bisectors of size O(Vng) in these graphs, both of which are optimal. As our main technical lemma, we prove an O(g/n) bound on the second smallest eigenvalue of the Laplacian of such graphs and show that this is tight, thereby resolving a conjecture of Spielman and Teng.

(cont.) While this lemma is essentially combinatorial in nature, its proof comes from continuous mathematics, drawing on the theory of circle packings and the geometry of compact Riemann surfaces. While the questions addressed in the two parts of the thesis are quite different, we show that a common methodology runs through their solutions. We believe that this methodology provides a powerful approach to the analysis of algorithms that will prove useful in a variety of broader contexts.