{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article8626","name":"Graph Motif Problems Parameterized by Dual","abstract":"Let G=(V,E) be a vertex-colored graph, where C is the set of colors used to color V. The Graph Motif (or GM) problem takes as input G, a multiset M of colors built from C, and asks whether there is a subset S subseteq V such that (i) G[S] is connected and (ii) the multiset of colors obtained from S equals M. The Colorful Graph Motif problem (or CGM) is a constrained version of GM in which M=C, and the List-Colored Graph Motif problem (or LGM) is the extension of GM in which each vertex v of V may choose its color from a list L(v) of colors.\r\n\r\nWe study the three problems GM, CGM and LGM, parameterized by l:=|V|-|M|. In particular, for general graphs, we show that, assuming the strong exponential-time hypothesis, CGM has no (2-epsilon)^l * |V|^{O(1)}-time algorithm, which implies that a previous algorithm, running in O(2^l\\cdot |E|) time is optimal. We also prove that LGM is W[1]-hard even if we restrict ourselves to lists of at most two colors. If we constrain the input graph to be a tree, then we show that, in contrast to CGM, GM can be solved in O(4^l *|V|) time but admits no polynomial kernel, while CGM can be solved in O(sqrt{2}^l + |V|) time and admits a polynomial kernel.","keywords":["NP-hard problem","subgraph problem","fixed-parameter algorithm","lowerbounds","kernelization"],"author":[{"@type":"Person","name":"Fertin, Guillaume","givenName":"Guillaume","familyName":"Fertin"},{"@type":"Person","name":"Komusiewicz, Christian","givenName":"Christian","familyName":"Komusiewicz"}],"position":7,"pageStart":"7:1","pageEnd":"7:12","dateCreated":"2016-06-27","datePublished":"2016-06-27","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Fertin, Guillaume","givenName":"Guillaume","familyName":"Fertin"},{"@type":"Person","name":"Komusiewicz, Christian","givenName":"Christian","familyName":"Komusiewicz"}],"copyrightYear":"2016","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.CPM.2016.7","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["http:\/\/dx.doi.org\/10.1089\/cmb.2009.0170","http:\/\/dx.doi.org\/10.1007\/978-3-662-44465-8","http:\/\/dx.doi.org\/10.1016\/j.jda.2014.03.002","http:\/\/www.lamsade.dauphine.fr\/~sikora\/pub\/GraphMotif-Resume.pdf"],"isPartOf":{"@type":"PublicationVolume","@id":"#volume6257","volumeNumber":54,"name":"27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016)","dateCreated":"2016-06-27","datePublished":"2016-06-27","editor":[{"@type":"Person","name":"Grossi, Roberto","givenName":"Roberto","familyName":"Grossi"},{"@type":"Person","name":"Lewenstein, Moshe","givenName":"Moshe","familyName":"Lewenstein"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article8626","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":"#volume6257"}}}