NDLI: On the Fine-Grained Complexity of Rainbow Coloring

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Lauri, Juho Socaa, Arkadiusz Kowalik, ukasz
Copyright Year	2018
Abstract	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 $k\ge 2$, there is no algorithm for Rainbow $k$-Coloring running in time $2^{o(n^{3/2})}$, unless the exponential time hypothesis 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. [J. Comb. Optim., 21 (2009), pp. 330--347], 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 fixed parameter tractable (FPT) when parameterized by $\|S\|$. We also study the Maximum 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 $k\ge 2$, extending the result of Ananth, Nasre, and Sarpatwar, in FSTTCS, LIPIcs, Schloss Dagstuhl--Leibniz-Zentum für Informatik, Dagstuhl, Germany, 2011, pp. 241--251. 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.
Sponsorship	Emil Aaltosen Säätiö. Narodowe Centrum Nauki
Starting Page	1672
Ending Page	1705
Page Count	34
File Format	PDF
ISSN	08954801
DOI	10.1137/16M1102690
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	3
Volume Number	32
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2018-07-17
Access Restriction	Subscribed
Subject Keyword	Computational difficulty of problems Nonnumerical algorithms Analysis of algorithms and problem complexity FPT algorithms graph coloring Graph algorithms Coloring of graphs and hypergraphs lower bounds computational complexity
Content Type	Text
Resource Type	Article
Subject	Mathematics

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