{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article9264","name":"Parameterized Algorithms for List K-Cycle","abstract":"The classic K-Cycle problem asks if a graph G, with vertex set V(G), has a simple cycle containing all vertices of a given set K subseteq V(G). In terms of colored graphs, it can be rephrased as follows: Given a graph G, a set K subset of V(G) and an injective coloring c from K to {1,2,...,|K|}, decide if G has a simple cycle containing each color in {1,2,...,|K|} (once). Another problem widely known since the introduction of color coding is {Colorful Cycle}. Given a graph G and a coloring c from V(G) to {1,2,...,k} for some natural number k, it asks if G has a simple cycle of length k containing each color in {1,2,...,k} (once). We study a generalization of these problems: Given a graph G, a set K subset of V(G), a list-coloring L from K to 2^{{1,2,...,k^*}} for some natural number k^* and a parameter k, List K-Cycle asks if one can assign a color to each vertex in K so that G would have a simple cycle (of arbitrary length) containing exactly k vertices from K with distinct colors.\r\n\r\nWe design a randomized algorithm for List K-Cycle running in time 2^kn^{O(1)} on an -vertex graph, matching the best known running times of algorithms for both K-Cycle and Colorful Cycle. Moreover, unless the Set Cover Conjecture is false, our algorithm is essentially optimal. We also study a variant of List K-Cycle that generalizes the classic Hamiltonicity problem, where one specifies the size of a solution. Our results integrate three related algebraic approaches, introduced by Bjorklund, Husfeldt and Taslaman (SODA'12), Bjorklund, Kaski and Kowalik (STACS'13), and Bjorklund (FOCS'10).","keywords":["Parameterized Complexity","K-Cycle","Colorful Path","k-Path"],"author":[{"@type":"Person","name":"Panolan, Fahad","givenName":"Fahad","familyName":"Panolan"},{"@type":"Person","name":"Zehavi, Meirav","givenName":"Meirav","familyName":"Zehavi"}],"position":22,"pageStart":"22:1","pageEnd":"22:15","dateCreated":"2016-12-10","datePublished":"2016-12-10","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Panolan, Fahad","givenName":"Fahad","familyName":"Panolan"},{"@type":"Person","name":"Zehavi, Meirav","givenName":"Meirav","familyName":"Zehavi"}],"copyrightYear":"2016","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.FSTTCS.2016.22","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6268","volumeNumber":65,"name":"36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)","dateCreated":"2016-12-10","datePublished":"2016-12-10","editor":[{"@type":"Person","name":"Lal, Akash","givenName":"Akash","familyName":"Lal"},{"@type":"Person","name":"Akshay, S.","givenName":"S.","familyName":"Akshay"},{"@type":"Person","name":"Saurabh, Saket","givenName":"Saket","familyName":"Saurabh"},{"@type":"Person","name":"Sen, Sandeep","givenName":"Sandeep","familyName":"Sen"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article9264","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":"#volume6268"}}}