David Karger
Rajeev Motwani
Abstract
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 n vertices with min{O(Δ1/3 log1/2 Δ log n), O(n1/4 log1/2 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 nontrivial 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(Δ1-2/k log1/2 Δ log n), O(n1−3/(k+1) log1/2 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 Lovász 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.
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Index Terms
- Approximate graph coloring by semidefinite programming
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Reviews
Adam Drozdek
The authors approach the coloring problem, which is known to be NP-hard, by finding an approximate optimum graph coloring. They relax the coloring problem by assigning unit vectors to graph vertices instead of assigning colors, and then requiring that, for two adjacent vertices i and j and their vectors v i and v j , the dot product v i , v j 1 k-1 . The authors first show that the vector chromatic number is between the clique number and the chromatic number. Next, they use the concept of semicoloring, that is, coloring at least half of the graph vertices, to show that an algorithm for semicoloring a graph can be used to color the graph with at most log n more colors than to semicolor it. They then present two algorithms that produce the coloring through a semicoloring of the graph. An assessment of the complexity of the two algorithms allows them to conclude that any vector k -colorable graph of n vertices and maximum degree D can be colored in randomized polynomial time with min O D 1-2k log 12 D log n ,O n 1-3 k+1 log 12 n colors. It is important to note that the approximation is given in terms of both n and D . The paper concludes by establishing a duality relationship between the semidefinite program solution given here and the Lova´sz j -function and the lower bound on the gap between the chromatic number and the semidefinite program solution.
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