On the Fine-Grained Complexity of Rainbow Coloring
Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Scientific › peer-review
Details
| Original language | English |
|---|---|
| Title of host publication | 24th Annual European Symposium on Algorithms (ESA 2016) |
| Editors | Piotr Sankowski, Christos Zaroliagis |
| Volume | 57 |
| ISBN (Electronic) | 978-3-95977-015-6 |
| DOIs | |
| Publication status | Published - 2016 |
| Publication type | A4 Article in a conference publication |
| Event | Annual European Symposium on Algorithms - Duration: 1 Jan 1900 → … |
Publication series
| Name | Leibniz International Proceedings in Informatics (LIPIcs) |
|---|---|
| Volume | 57 |
| ISSN (Electronic) | 1868-8969 |
Conference
| Conference | Annual European Symposium on Algorithms |
|---|---|
| Period | 1/01/00 → … |
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 >= 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.
Keywords
- graph coloring, computational complexity, lower bounds, exponential time hypothesis, FPT algorithms