approx - Introduction to Approximation Algorithms Lecture...

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Unformatted text preview: Introduction to Approximation Algorithms Lecture 12: Mar 1 NP-completeness We have seen many polynomial time solvable optimization problems. e.g. maximum matching, min-cost 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. NP-completeness NP (Non-deterministic 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. NP-completeness 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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approx - Introduction to Approximation Algorithms Lecture...

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