License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2017.60
URN: urn:nbn:de:0030-drops-80990
URL: https://drops.dagstuhl.de/opus/volltexte/2017/8099/
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### Fine-Grained Complexity of Rainbow Coloring and its Variants

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### Abstract

Consider a graph G and an edge-coloring c_R:E(G) \rightarrow [k]. A rainbow path between u,v \in V(G) is a path P from u to v such that for all e,e' \in E(P), where e \neq e' we have c_R(e) \neq c_R(e'). In the Rainbow k-Coloring problem we are given a graph G, and the objective is to decide if there exists c_R: E(G) \rightarrow [k] such that for all u,v \in V(G) there is a rainbow path between u and v in G. Several variants of Rainbow k-Coloring have been studied, two of which are defined as follows. The Subset Rainbow k-Coloring takes as an input a graph G and a set S \subseteq V(G) \times V(G), and the objective is to decide if there exists c_R: E(G) \rightarrow [k] such that for all (u,v) \in S there is a rainbow path between u and v in G. The problem Steiner Rainbow k-Coloring takes as an input a graph G and a set S \subseteq V(G), and the objective is to decide if there exists c_R: E(G) \rightarrow [k] such that for all u,v \in S there is a rainbow path between u and v in G. In an attempt to resolve open problems posed by Kowalik et al. (ESA 2016), we obtain the following results.

- For every k \geq 3, Rainbow k-Coloring does not admit an algorithm running in time 2^{o(|E(G)|)}n^{O(1)}, unless ETH fails.

- For every k \geq 3, Steiner Rainbow k-Coloring does not admit an algorithm running in time 2^{o(|S|^2)}n^{O(1)}, unless ETH fails.

- Subset Rainbow k-Coloring admits an algorithm running in time 2^{\OO(|S|)}n^{O(1)}. This also implies an algorithm running in time 2^{o(|S|^2)}n^{O(1)} for Steiner Rainbow k-Coloring, which matches the lower bound we obtain.

### BibTeX - Entry

@InProceedings{agrawal:LIPIcs:2017:8099,
author =	{Akanksha Agrawal},
title =	{{Fine-Grained Complexity of Rainbow Coloring and its Variants}},
booktitle =	{42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)},
pages =	{60:1--60:14},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-046-0},
ISSN =	{1868-8969},
year =	{2017},
volume =	{83},
editor =	{Kim G. Larsen and Hans L. Bodlaender and Jean-Francois Raskin},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address =	{Dagstuhl, Germany},
URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/8099},
URN =		{urn:nbn:de:0030-drops-80990},
doi =		{10.4230/LIPIcs.MFCS.2017.60},
annote =	{Keywords: Rainbow Coloring, Lower bound, ETH, Fine-grained Complexity}
}


 Keywords: Rainbow Coloring, Lower bound, ETH, Fine-grained Complexity Collection: 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017) Issue Date: 2017 Date of publication: 01.12.2017

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