Approximation schemes for NP-hard geometric optimization problems: a survey

NP-hard geometric optimization problems arise in many disciplines. Perhaps the most famous one is the traveling salesman problem (TSP): given n nodes in<2 (more generally, in 1. All these are subcases of the more general notion of a geometric norm or Minkowski norm. We will refer to the version of the problem with a general geometric norm as geometric TSP. Some other NP-hard geometric optimization problems are Minimum Steiner Tree (“Given n points, find the smallest network connecting them,”), kTSP(“Given n points and a number k, find the shortest salesman tour that visits k points”), k-MST (“Given n points and a number k, find the shortest tree that contains k points”), vehicle routing, degree restricted minimum spanning tree, etc. Thus if P≠NP, as is widely conjectured, we cannot design polynomial time algorithms to solve these problems optimally. However, we might be able to design approximation algorithms: algorithms that compute near-optimal solutions in polynomial time for every problem instance. For α ≥ 1 we say that an algorithm approximates the problem within a factor α if it computes, for every instance I, a solution of cost at most α ·OPT(I), where OPT(I) is the cost of the optimum solution for I. (The preceding definition is for minimization problems; for maximization problems α ≤ 1.) Sometimes we use the shortened name “α-approximation algorithm.” Bern and Eppstein [14] give an excellent survey circa 1995 of approximation algorithms for geometric problems. The current survey will concentrate on developments subsequent to 1995, many of which followed the author’s discovery, in 1996, of a polynomial time approximation scheme or “PTAS” for geometric TSP in constant dimensions. (By “constant dimensions” we mean that we fix the dimension d and consider asymptotic complexity as we increase n, the number of nodes.) A PTAS is an “ultimate” approximation ∗35 Olden St, Princeton NJ 08544. aroracs.princeton.edu. Supported by a David and Lucile Packard Fellowship and NSF grant CCR-0098180