Approximating k-center in planar graphs

Abstract

We consider variants of the metric k-center problem. Imagine that you must choose locations for k firehouses in a city so as to minimize the maximum distance of a house from the nearest firehouse. An instance is specified by a graph with arbitrary nonnegative edge lengths, a set of vertices that can serve as firehouses (i.e., centers) and a set of vertices that represent houses. For general graphs, this problem is exactly equivalent to the metric k-center problem, which is APX-hard. We give a polynomial-time bicriteria approximation scheme when the input graph is a planar graph. We also give polynomial-time bicriteria approximation schemes for several generalizations: if, instead of all houses, we wish to cover a specified proportion of the houses; if the candidate locations for firehouses have rental costs and we wish to minimize not the number of firehouses but the sum of their rental costs; and if the input graph is not planar but is of bounded genus. Copyright © 2014 by the Society for Industrial and Applied Mathematics.