In SocG 2009, Mustafa and Ray showed that a local search algorithm is, somewhat surprisingly, a PTAS for certain geometric set cover/hitting set problems. In this context, a PTAS is a scheme that for any eps > 0, yields an algorithm that runs in polynomial time and returns a solution whose size is at most (1 + eps) times that of the optimal set cover/hitting set. A similar PTAS was presented by Chan and Har-Peled at the same conference for the largest independent set problem in certain geometric intersection graphs.
We will discuss a hitting set result of Mustafa and Ray and, time permitting, an application of their machinery to a terrain guarding problem by Gibson, Kanade, Krohn, and the speaker.
