Approximate graph coloring by semidefinite programming
Karger, D.
Motwani, R.
Sudan, M.
Stanford Univ., CA;
This paper appears in: Foundations of Computer Science, 1994 Proceedings., 35th Annual Symposium on
Publication Date: 20-22 Nov 1994
On page(s): 2-13
Meeting Date: 11/20/1994 - 11/22/1994
Location: Santa Fe, NM, USA
ISBN: 0-8186-6580-7
References Cited: 40
INSPEC Accession Number: 4865010
Digital Object Identifier: 10.1109/SFCS.1994.365710
Current Version Published: 2002-08-06
Abstract
We consider the problem of coloring k-colorable graphs with the
fewest possible colors. We give a randomized polynomial time algorithm
which colors a 3-colorable graph on n vertices with min
{O(Δ1/3log4/3Δ), O(n1/4 log
n)} colors where Δ is the maximum degree of any vertex. Besides
giving the best known approximation ratio in terms of n, this marks the
first non-trivial approximation result as a function of the maximum
degree Δ. This result can be generalized to k-colorable graphs to
obtain a coloring using min {O˜(Δ1-2k/),
O˜(n1-3(k+1/))} 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 Lovasz ϑ-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 ϑ-function
Index
Terms
Available to subscribers and IEEE members.
References
Available to subscribers and IEEE members.
Citing Documents
Available to subscribers and IEEE members.
Cart
