Abstract

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: ffl A simple, unified approach to a number of problems on multicommodity flows, including the Leighton--Rao Theorem [36] and some of its extensions. We solve an open question in this area, showing that the max--flow vs. min--c...

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