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Author Karger, David ♦ Motwani, Rajeev ♦ Sudan, Madhu
Source ACM Digital Library
Content type Text
Publisher Association for Computing Machinery (ACM)
File Format PDF
Copyright Year ©1998
Language English
Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Subject Keyword NP-completeness ♦ Approximation algorithms ♦ Chromatic number ♦ Graph coloring ♦ Randomized algorithms
Abstract We consider the problem of coloring $\textit{k}-colorable$ graphs with the fewest possible colors. We present a randomized polynomial time algorithm that colors a 3-colorable graph on $\textit{n}$ vertices with $min{\textit{O}(Δ1/3$ log1/2 Δ log $\textit{n}),$ $\textit{O}(\textit{n}1/4$ log1/2 $\textit{n})}$ colors where Δ is the maximum degree of any vertex. Besides giving the best known approximation ratio in terms of $\textit{n},$ this marks the first nontrivial approximation result as a function of the maximum degree Δ. This result can be generalized to $\textit{k}-colorable$ graphs to obtain a coloring using $min{\textit{O}(Δ1-2/\textit{k}$ log1/2 Δ log $\textit{n}),$ $\textit{O}(\textit{n}1™3/(\textit{k}+1)$ log1/2 $\textit{n})}$ colors. Our results are inspired by the recent work of Goemans and Williamson who used an algorithm for semidefinite optimization problems, which generalize linear programs, to obtain improved approximations for the MAX CUT and MAX 2-SAT problems. An intriguing outcome of our work is a duality relationship established between the value of the optimum solution to our semidefinite program and the Lovász &thgr;-function. We show lower bounds on the gap between the optimum solution of our semidefinite program and the actual chromatic number; by duality this also demonstrates interesting new facts about the &thgr;-function.
ISSN 00045411
Age Range 18 to 22 years ♦ above 22 year
Educational Use Research
Education Level UG and PG
Learning Resource Type Article
Publisher Date 1998-03-01
Publisher Place New York
e-ISSN 1557735X
Journal Journal of the ACM (JACM)
Volume Number 45
Issue Number 2
Page Count 20
Starting Page 246
Ending Page 265

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Source: ACM Digital Library