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Communication Dans Un Congrès Année : 2015

Improved Local Search for Geometric Hitting Set

Norbert Bus
Shashwat Garg
  • Fonction : Auteur
  • PersonId : 969305
Nabil Mustafa
Saurabh Ray
  • Fonction : Auteur
  • PersonId : 969306
NYU

Résumé

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, the 30-year quest for a PTAS, starting from the seminal work of Hochbaum [19], was finally achieved in 2010. Surprisingly, the algorithm to achieve the PTAS is simple: 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 hopeless running times; e.g., a 3-approximation for the geometric hitting takes more than n 66 time [15] for the geometric hitting set problem for disks in the plane! 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 one of the fundamental problems, hitting sets for disks in the plane. We present tight approximation bounds for (3, 2)-local search 1 and give an (8 + algorithm with running time O(n 2.34); the previous-best result achieving a similar approximation ratio gave a 10-approximation in time O(n 15) – 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 better 2. Similarly (4, 3)-local search gives a 5-approximation for all these problems.

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Dates et versions

hal-01188990 , version 1 (01-09-2015)

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  • HAL Id : hal-01188990 , version 1

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Norbert Bus, Shashwat Garg, Nabil Mustafa, Saurabh Ray. Improved Local Search for Geometric Hitting Set. Proc. of the 32st International Symposium on Theoretical Aspects of Computer Science (STACS), 2015, Munich, Germany. ⟨hal-01188990⟩
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