cs601_homework8

cs601_homework8 - Algorithms-Complexity Homework-8 Prateek...

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Algorithms-Complexity Homework-8 Prateek Sharma - 09305910 November 10, 2009 1 Problem 1 - 3-Coloring We use similar techniques like we did for the MAX-3-SAT randomized algorithm. Pick any vertex and assign a color randomly. Since there are 3 colors, the probability is 2 / 3 that a random assignment of colors to these two vertices is safe(legal). That is the 2 colors need to be different. Let X be event that the coloring of 2 vertices is legal (diff colored). Let Y = ¯ X be the event that they are assigned same color. Pr ( Y ) = 1 / 3. Hence Pr ( X ) = 1 - Pr ( ¯ X ). Each edge has the probability of 2 / 3 that its associated vertices are safely colored. Consider OPT, the optimal 3-coloring algorithm. Let OPT be the number of correctly colored vertices. Then , OPT E . The randomized algorithm described above. Let the number of edges cor- rectly colored (by that we really mean that the endpoints are differently colored) be N . Then expected number of edges returned by the randomized algorithm
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cs601_homework8 - Algorithms-Complexity Homework-8 Prateek...

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