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On the Fine-Grained Complexity of Rainbow Coloring

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Original languageEnglish
Title of host publication24th Annual European Symposium on Algorithms (ESA 2016)
EditorsPiotr Sankowski, Christos Zaroliagis
ISBN (Electronic)978-3-95977-015-6
Publication statusPublished - 2016
Publication typeA4 Article in a conference publication
EventAnnual European Symposium on Algorithms -
Duration: 1 Jan 1900 → …

Publication series

NameLeibniz International Proceedings in Informatics (LIPIcs)
ISSN (Electronic)1868-8969


ConferenceAnnual European Symposium on Algorithms
Period1/01/00 → …


The Rainbow k-Coloring problem asks whether the edges of a given graph can be colored in k colors so that every pair of vertices is connected by a rainbow path, i.e., a path with all edges of different colors. Our main result states that for any k >= 2, there is no algorithm for Rainbow k-Coloring running in time 2^{o(n^{3/2})}, unless ETH fails. Motivated by this negative result we consider two parameterized variants of the problem. In the Subset Rainbow k-Coloring problem, introduced by Chakraborty et al. [STACS 2009, J. Comb. Opt. 2009], we are additionally given a set S of pairs of vertices and we ask if there is a coloring in which all the pairs in S are connected by rainbow paths. We show that Subset Rainbow k-Coloring is FPT when parameterized by |S|. We also study Subset Rainbow k-Coloring problem, where we are additionally given an integer q and we ask if there is a coloring in which at least q anti-edges are connected by rainbow paths. We show that the problem is FPT when parameterized by q and has a kernel of size O(q) for every k >= 2, extending the result of Ananth et al. [FSTTCS 2011]. We believe that our techniques used for the lower bounds may shed some light on the complexity of the classical Edge Coloring problem, where it is a major open question if a 2^{O(n)}-time algorithm exists.


  • graph coloring, computational complexity, lower bounds, exponential time hypothesis, FPT algorithms

Publication forum classification

Field of science, Statistics Finland