Approximation algorithms for geometric problems
Marshall W. Bern, David Eppstein · 1996
INTRODUCTION 8.1 This chapter surveys approximation algorithms for hard geometric problems. The problems we consider typically take inputs that are point sets or polytopes in two- or three-dimensional space, and seek optimal constructions, (which may be trees, paths, or polytopes). We limit attention to problems for which no polynomial-time exact algorithms are known, and concentrate on bounds for worst-case approximation ratios, especially bounds that depend intrinsically on geometry. We illustrate our intentions with two well-known problems. Given a finite set of points S in the plane, the Euclidean traveling salesman problem asks for the shortest tour of S. Christofides' algorithm achieves approximation ratio 3 2 for this problem, meaning that it always computes a tour of length at most three-halves the length of the optimal tour. This bound depends only on the triangle inequality, so Christofides' algorit