The geometry of graphs and some of its algorithmic applications

In this paper we explore some implications of viewing graphs as geometric objects. This approach offers a new perspective on a number of graph-theoretic and algorithmic problems. There are several ways to model graphs geometrically and our main concern here is with geometric representations that respect the metric of the (possibly weighted) graph. Given a graph G we map its vertices to a normed space in an attempt to (i) keep down the dimension of the host space, and (ii) guarantee a small distortion, i.e., make sure that distances between vertices in G closely match the distances between their geometric images. In this paper we develop efficient algorithms for embedding graphs low-dimensionally with a small distortion. Further algorithmic applications include: • A simple, unified approach to a number of problems on multicommodity flows, including the Leighton-Rao Theorem [37] and some of its extensions. We solve an open question in this area, showing that the max-flow vs. min-cut gap in the k-commodities problem is O(log k). Our new deterministic polynomial-time algorithm finds a (nearly tight) cut meeting this bound. • For graphs embeddable in low-dimensional spaces with a small distortion, we can find low-diameter decompositions (in the sense of [7] and [43]). The parameters of the decomposition depend only on the dimension and the distortion and not on the size of the graph. • In graphs embedded this way, small balanced separators can be found efficiently. Given faithful low-dimensional representations of statistical data, it is possible to obtain meaningful and efficient clustering. This is one of the most basic tasks in pattern-recognition. For the (mostly heuristic) methods used in the practice of pattern-recognition, see [20], especially chapter 6. Our studies of multicommodity flows also imply that every embedding of (the metric of) an n-vertex, constant-degree expander into a Euclidean space (of any dimension) has distortion Ω(log n). This result is tight, and closes a gap left open by Bourgain [12].