Improved Local Search for Geometric Hitting Set

Over the past several decades there has been steady progress towards the goal of polynomial-time approximation schemes (PTAS) for fundamental geometric combinatorial optimization problems. A foremost example is the geometric hitting set problem: given a set P of points and a set D of geometric objects, compute the minimum-sized subset of P that hits all objects in D. For the case where D is a set of disks in the plane, a PTAS was finally achieved in 2010, with a surprisingly simple algorithm based on local-search. Since then, local-search has turned out to be a powerful algorithmic approach towards achieving good approximation ratios for geometric problems (for geometric independent-set problem, for dominating sets, for the terrain guarding problem and several others). Unfortunately all these algorithms have the same limitation: local search is able to give a PTAS, but with large running times. That leaves open the question of whether a better understanding – both combinatorial and algorithmic – of local search and the problem can give a better approximation ratio in a more reasonable time. In this paper, we investigate this question for hitting sets for disks in the plane. We present tight approximation bounds for (3, 2)local search and give an (8 + )-approximation algorithm with expected running time Õ(n2.34); the previous-best result achieving a similar approximation ratio gave a 10-approximation in time O(n15) – that too just for unit disks. The techniques and ideas generalize to (4, 3) local search. Furthermore, as mentioned earlier, local-search has been used for several other geometric optimization problems; for all these problems our results show that (3, 2) local search gives an 8-approximation and no better1. Similarly (4, 3)-local search gives a 5-approximation for all these problems. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems