## On the Fine-Grained Complexity of Rainbow Coloring

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Scientific › peer-review

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**On the Fine-Grained Complexity of Rainbow Coloring.** / Kowalik, Lukasz; Lauri, Juho; Socala, Arkadiusz.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Scientific › peer-review

### Harvard

*24th Annual European Symposium on Algorithms (ESA 2016).*vol. 57, Leibniz International Proceedings in Informatics (LIPIcs), vol. 57, Annual European Symposium on Algorithms, 1/01/00. https://doi.org/10.4230/LIPIcs.ESA.2016.58

### APA

*24th Annual European Symposium on Algorithms (ESA 2016)*(Vol. 57). (Leibniz International Proceedings in Informatics (LIPIcs); Vol. 57). https://doi.org/10.4230/LIPIcs.ESA.2016.58

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TY - GEN

T1 - On the Fine-Grained Complexity of Rainbow Coloring

AU - Kowalik, Lukasz

AU - Lauri, Juho

AU - Socala, Arkadiusz

N1 - JUFOID=79091

PY - 2016

Y1 - 2016

N2 - 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.

AB - 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.

KW - graph coloring

KW - computational complexity

KW - lower bounds

KW - exponential time hypothesis

KW - FPT algorithms

U2 - 10.4230/LIPIcs.ESA.2016.58

DO - 10.4230/LIPIcs.ESA.2016.58

M3 - Conference contribution

VL - 57

T3 - Leibniz International Proceedings in Informatics (LIPIcs)

BT - 24th Annual European Symposium on Algorithms (ESA 2016)

A2 - Sankowski, Piotr

A2 - Zaroliagis, Christos

ER -