TY - GEN
T1 - Breaking the 2nBarrier for 5-Coloring and 6-Coloring
AU - Zamir, Or
N1 - Publisher Copyright: © 2021 Or Zamir.
PY - 2021/7/1
Y1 - 2021/7/1
N2 - 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 a 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-∈) n) time for some ∈ > 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 - ∈) n) time, for some ∈ = ∈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.
AB - 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 a 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-∈) n) time for some ∈ > 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 - ∈) n) time, for some ∈ = ∈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.
KW - Algorithms
KW - Graph algorithms
KW - Graph coloring
UR - http://www.scopus.com/inward/record.url?scp=85115332643&partnerID=8YFLogxK
U2 - https://doi.org/10.4230/LIPIcs.ICALP.2021.113
DO - https://doi.org/10.4230/LIPIcs.ICALP.2021.113
M3 - منشور من مؤتمر
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021
A2 - Bansal, Nikhil
A2 - Merelli, Emanuela
A2 - Worrell, James
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021
Y2 - 12 July 2021 through 16 July 2021
ER -