License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2021.113
URN: urn:nbn:de:0030-drops-141825
URL: https://drops.dagstuhl.de/opus/volltexte/2021/14182/
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Zamir, Or

Breaking the 2ⁿ Barrier for 5-Coloring and 6-Coloring

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LIPIcs-ICALP-2021-113.pdf (0.9 MB)


Abstract

The coloring problem (i.e., computing the chromatic number of a graph) can be solved in O^*(2ⁿ) time, as shown by Björklund, Husfeldt and Koivisto in 2009. For k = 3,4, better algorithms are known for the k-coloring problem. 3-coloring can be solved in O(1.33ⁿ) time (Beigel and Eppstein, 2005) and 4-coloring can be solved in O(1.73ⁿ) time (Fomin, Gaspers and Saurabh, 2007). Surprisingly, for k > 4 no improvements over the general O^*(2ⁿ) are known. We show that both 5-coloring and 6-coloring can also be solved in O((2-ε) ⁿ) time for some ε > 0. As a crucial step, we obtain an exponential improvement for computing the chromatic number of a very large family of graphs. In particular, for any constants Δ,α > 0, the chromatic number of graphs with at least α⋅ n vertices of degree at most Δ can be computed in O((2-ε) ⁿ) time, for some ε = ε_{Δ,α} > 0. This statement generalizes previous results for bounded-degree graphs (Björklund, Husfeldt, Kaski, and Koivisto, 2010) and graphs with bounded average degree (Golovnev, Kulikov and Mihajlin, 2016). We generalize the aforementioned statement to List Coloring, for which no previous improvements are known even for the case of bounded-degree graphs.

BibTeX - Entry

@InProceedings{zamir:LIPIcs.ICALP.2021.113,
  author =	{Zamir, Or},
  title =	{{Breaking the 2ⁿ Barrier for 5-Coloring and 6-Coloring}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{113:1--113:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/14182},
  URN =		{urn:nbn:de:0030-drops-141825},
  doi =		{10.4230/LIPIcs.ICALP.2021.113},
  annote =	{Keywords: Algorithms, Graph Algorithms, Graph Coloring}
}

Keywords: Algorithms, Graph Algorithms, Graph Coloring
Collection: 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)
Issue Date: 2021
Date of publication: 02.07.2021


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