C H A P T E R 8 APPROXIMATION ALGORITHMS FOR GEOMETRIC PROBLEMS Marshall Bern David Eppstein This chapter discusses approximation algorithms for hard geometric problems. We cover three well-known shortest network problems—traveling salesman, Steiner tree, and minimum weight triangulation—along with an assortment of problems in areas such as clustering and surface approximation. INTRODUCTION 8.1 This chapter surveys approximation algorithms for hard geometric problems. The prob- lems 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 al- gorithms 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’ algorithm works equally well in any metric space, even the finite metric space induced by the shortest-path distance on a network (edge-weighted graph). The Steiner tree problem asks for the shortest tree that includes S in its vertex set. Approximation ratios for this problem depend quite intimately on geometry. For 296