Approximation Schemes for Geometric Coverage Problems

In their seminal work, Mustafa and Ray [25] showed that a wide class of geometric set cover (SC) problems admit a PTAS via local search, which appears to be the most powerful approach known for such problems. Their result applies if a naturally defined “exchange graph” for two feasible solutions is planar and is based on subdividing this graph via a planar separator theorem due to Frederickson [16]. Obtaining similar results for the related maximum k-coverage problem (MC) seems challenging due to the hard cardinality constraint. In fact, in a subsequent work, Badanidiyuru, Kleinberg, and Lee [2] study geometric MC. They obtain fixed-parameter approximation schemes for MC instances with bounded VC dimension, but the running times are exponential in k. In this work, we overcome the above-mentioned technical hurdles by proposing a colored version of the planar separator theorem that might be of independent interest. The resulting subdivision approximates locally in each part the global distribution of the colors. This allows us to obtain a PTAS (with running time polynomial in k) for any “planarizable” instance of MC and thus essentially for all cases where the corresponding SC instance can be tackled via the approach of Mustafa and Ray. As a corollary, we resolve an open question by Badanidiyuru, Kleinberg, and Lee regarding real half spaces in dimension three.