Approximating geometrical graphs via `spanners' and `banyans'

Abstract

The main result of this paper is an improvement of Arora's method to find (1+ε) approximations for geometric NP-hard problems including the Euclidean Traveling Salesman Problem and the Euclidean Steiner Minimum Tree problems. For fixed dimension d and ε, our algorithms run in O(N log N) time. An interesting byproduct of our work is the definition and construction of banyans, a generalization of graph spanners. A (1+ε)-banyan for a set of points A is a set of points A′ and line segments S with endpoints in A in union of all the sets A′ such that a 1+ε optimal Steiner Minimum Tree for any subset of A is contained in S. We give a construction for banyans such that the total length of the line segments in S is within a constant factor of the size of the minimum spanning tree of A when ε and d are fixed. In this abbreviated paper, we only provide proofs of these results in two dimensions. The full paper on WDS's web page (http://www.neci.nj.nec.com/homepages/wds, click `NECI technical reports') extends the techniques to higher dimensions, proves some new facts and clarifies some old facts about spanners, and also gives approximation algorithms for minimum matching, edge cover, rectilinear Steiner minimum tree, and minimum 2-matching.