Approximation algorithms for NP-hard optimization problems
Algorithms and theory of computation handbook
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A coloring of a graph is an assignment of colors to the vertices so that no two adjacent vertices are given the same color. This thesis describes polynomial-time algorithms for coloring k-colorable graphs with as few additional colors possible, focusing on 3-colorable graphs. For the worst-case problem, we present an algorithm that improves upon previous bounds for the number of colors used and colors any 3-colorable n-node graph with just over n 3/8 colors, thus breaking an ******** barrier. In addition, we improve bounds for coloring k-colorable graphs and consider a semi-random graph model (based upon the semi-random source of Santha and Vazirani) that provides a smooth transition between the worst-case and random scenarios. The randomness of the graph in the semi-random model is determined by a noise rate, and we show an algorithm that for a wide variety of noise rates will k-color semi-random k-colorable graphs. Finally, using a basic technique developed by Berman and Schnitger, we show that if there do not exist polynomial-time algorithms to color k-colorable graphs with O (log n) colors, then the largest Independent Set in a graph (or equivalently the largest Clique) cannot be approximated to within a factor of for any constant