{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article10315","name":"On the Parameterized Complexity of Red-Blue Points Separation","abstract":"We study the following geometric separation problem: Given a set R of red points and a set B of blue points in the plane, find a minimum-size set of lines that separate R from B. We show that, in its full generality, parameterized by the number of lines k in the solution, the problem is unlikely to be solvable significantly faster than the brute-force n^{O(k)}-time algorithm, where n is the total number of points.\r\n Indeed, we show that an algorithm running in time f(k)n^{o(k\/log k)}, for any computable function f, would disprove ETH. Our reduction crucially relies on selecting lines from a set with a large number of different slopes (i.e., this number is not a function of k).\r\n\r\nConjecturing that the problem variant where the lines are required to be axis-parallel is FPT in the number of lines, we show the following preliminary result. \r\nSeparating R from B with a minimum-size set of axis-parallel lines is FPT in the size of either set, and can be solved in time O^*(9^{|B|}) (assuming that B is the smallest set).","keywords":["red-blue points separation","geometric problem","W[1]-hardness","FPT algorithm","ETH-based lower bound"],"author":[{"@type":"Person","name":"Bonnet, \u00c9douard","givenName":"\u00c9douard","familyName":"Bonnet"},{"@type":"Person","name":"Giannopoulos, Panos","givenName":"Panos","familyName":"Giannopoulos"},{"@type":"Person","name":"Lampis, Michael","givenName":"Michael","familyName":"Lampis"}],"position":8,"pageStart":"8:1","pageEnd":"8:13","dateCreated":"2018-03-02","datePublished":"2018-03-02","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Bonnet, \u00c9douard","givenName":"\u00c9douard","familyName":"Bonnet"},{"@type":"Person","name":"Giannopoulos, Panos","givenName":"Panos","familyName":"Giannopoulos"},{"@type":"Person","name":"Lampis, Michael","givenName":"Michael","familyName":"Lampis"}],"copyrightYear":"2018","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.IPEC.2017.8","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["http:\/\/dx.doi.org\/10.4230\/LIPIcs.ESA.2016.19","http:\/\/dx.doi.org\/10.1016\/j.dam.2003.11.014"],"isPartOf":{"@type":"PublicationVolume","@id":"#volume6292","volumeNumber":89,"name":"12th International Symposium on Parameterized and Exact Computation (IPEC 2017)","dateCreated":"2018-03-02","datePublished":"2018-03-02","editor":[{"@type":"Person","name":"Lokshtanov, Daniel","givenName":"Daniel","familyName":"Lokshtanov"},{"@type":"Person","name":"Nishimura, Naomi","givenName":"Naomi","familyName":"Nishimura"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article10315","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":"#volume6292"}}}