Title from Practice to Theory; Approximation Schemes for Clustering and Network Design

Abstract What are the performance guarantees of the algorithms used in practice for clustering and network design problems? We answer this question by showing that the standard local search algorithm returns a nearly-optimal solution for low-dimensional Euclidean instances of the traveling salesman problem, Steiner tree, k-median and k-means. The result also extends to the case of graphs excluding a fixed minor for k-median and k-means. The problem of finding a polynomial-time approximation scheme for instances of the k-means problem consisting of points lying in a low-dimensional Euclidean space or graphs excluding a fixed minor is a 15-year old open problem. In this work, we solve this problem by showing that local search is a polynomial-time approximation scheme for those instances. Furthermore, we show that for three recent characterizations of clustering instances stemming from machine learning and data analysis, local search returns a nearly-optimal solution. One of the key ingredients that makes local search efficient is the existence of “cheap separators” in the instances. In the last part of this work, we show how to compute a more structured separator tailored for network design problems. This yields a general framework for obtaining approximation schemes for connectivity and domination problems in weighted planar graphs. It implies the first polynomial-time approximation schemes for weighted versions of tree and tour cover, connected vertex cover, connected dominating set, and feedback vertex set in planar graphs.