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# Fine-Grained Complexity of Rainbow Coloring and its Variants

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LIPIcs.MFCS.2017.60.pdf
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## Cite As

Akanksha Agrawal. Fine-Grained Complexity of Rainbow Coloring and its Variants. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 60:1-60:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.MFCS.2017.60

## 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.
##### Keywords
• Rainbow Coloring
• Lower bound
• ETH
• Fine-grained Complexity

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## References

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