{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article7717","name":"Improved Local Search for Geometric Hitting Set","abstract":"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).\r\n\r\nUnfortunately 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+\\epsilon)-approximation algorithm with expected running time \u02dcO(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 \\footnote{This is assuming the use of the standard framework. Improvement of the approximation factor by using additional properties specific to the problem may be possible.}. Similarly (4,3)-local search gives a 5-approximation for all these problems.","keywords":["hitting sets","Delaunay triangulation","local search","disks","geometric algorithms"],"author":[{"@type":"Person","name":"Bus, Norbert","givenName":"Norbert","familyName":"Bus"},{"@type":"Person","name":"Garg, Shashwat","givenName":"Shashwat","familyName":"Garg"},{"@type":"Person","name":"Mustafa, Nabil H.","givenName":"Nabil H.","familyName":"Mustafa"},{"@type":"Person","name":"Ray, Saurabh","givenName":"Saurabh","familyName":"Ray"}],"position":18,"pageStart":184,"pageEnd":196,"dateCreated":"2015-02-26","datePublished":"2015-02-26","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Bus, Norbert","givenName":"Norbert","familyName":"Bus"},{"@type":"Person","name":"Garg, Shashwat","givenName":"Shashwat","familyName":"Garg"},{"@type":"Person","name":"Mustafa, Nabil H.","givenName":"Nabil H.","familyName":"Mustafa"},{"@type":"Person","name":"Ray, Saurabh","givenName":"Saurabh","familyName":"Ray"}],"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.STACS.2015.184","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6233","volumeNumber":30,"name":"32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)","dateCreated":"2015-02-26","datePublished":"2015-02-26","editor":[{"@type":"Person","name":"Mayr, Ernst W.","givenName":"Ernst W.","familyName":"Mayr"},{"@type":"Person","name":"Ollinger, Nicolas","givenName":"Nicolas","familyName":"Ollinger"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article7717","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6233"}}}