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Unformatted text preview: Introduction to Approximation Algorithms Lecture 12: Mar 1 NPcompleteness We have seen many polynomial time solvable optimization problems. e.g. maximum matching, mincost flow, minimum cut, etc. However, there are much more optimization problems that we do not know how to solve in polynomial time. e.g. traveling salesman, graph colorings, maximum independent set, set cover, maximum clique, maximum cut, minimum Steiner tree, satisfiability, etc. Vertex Cover Vertex cover : a subset of vertices which “ covers ” every edge. An edge is covered if one of its endpoint is chosen. The Minimum Vertex Cover Problem : Find a vertex cover with minimum number of vertices. NPcompleteness NP (Nondeterministic polynomial time): A class of decision problems whose solutions can be “verified” in polynomial time. For each “yes” instance, there is a proof that can be checked in polynomial time. Decision problem for vertex cover: Is there a vertex cover of size at most k? Examples of NP problems: traveling salesman, graph colorings, maximum independent set, set cover, maximum clique, maximum cut, minimum Steiner tree, satisfiability, etc. NPcompleteness For each “yes” instance of the vertex cover problem, the proof is just a set of k vertices which cover all the edges....
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 Fall '11
 GraceGraham
 Computational complexity theory, vertex cover, NPcomplete problems, Boolean satisfiability problem

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