1,168 research outputs found
Approximate Graph Coloring by Semidefinite Programming
We consider the problem of coloring k-colorable graphs with the fewest
possible colors. We present a randomized polynomial time algorithm that colors
a 3-colorable graph on vertices with min O(Delta^{1/3} log^{1/2} Delta log
n), O(n^{1/4} log^{1/2} n) colors where Delta 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 Delta. This result can be generalized to k-colorable graphs to obtain a
coloring using min O(Delta^{1-2/k} log^{1/2} Delta log n), O(n^{1-3/(k+1)}
log^{1/2} 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 Lovasz theta-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 theta-function
A polyhedral approach for the Equitable Coloring Problem
In this work we study the polytope associated with a 0,1-integer programming
formulation for the Equitable Coloring Problem. We find several families of
valid inequalities and derive sufficient conditions in order to be
facet-defining inequalities. We also present computational evidence that shows
the efficacy of these inequalities used in a cutting-plane algorithm
- ā¦