2026-07-22T05:43:27Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:6400 2024-03-06T10:38:10Z ddc:004 open_access

On the Fine-Grained Complexity of Rainbow Coloring Kowalik, Lukasz Lauri, Juho Socala, Arkadiusz graph coloring computational complexity lower bounds exponential time hypothesis FPT algorithms 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Lukasz Kowalik and Juho Lauri and Arkadiusz Socala 2016 Is Part Of LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ESA.2016.58 urn:nbn:de:0030-drops-64001 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.58 eng https://creativecommons.org/licenses/by/3.0/legalcode