L15-cover - Introduction to Approximation Algorithms...

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1 Introduction to Approximation Algorithms Lecture 15: Mar 5
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2 NP-completeness Do your best then.
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3 Different Approaches Special graph classes e.g. vertex cover in bipartite graphs, perfect graphs. Fast exact algorithms, fixed parameter algorithms find a vertex cover of size k efficiently for small k. Average case analysis find an algorithm which works well on average. Approximation algorithms find an algorithm which return solutions that are guaranteed to be close to an optimal solution.
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4 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.
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5 Approximation Algorithms Constant factor approximation algorithms : SOL <= cOPT for some constant c. Key: provably close to optimal. Let OPT be the value of an optimal solution, and let SOL be the value of the solution that our algorithm returned.
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6 Vertex Cover: Greedy Algorithm 1 Idea: Keep finding a vertex which covers the maximum number of edges. Greedy Algorithm 1: 1. Find a vertex v with maximum degree. 2. Add v to the solution and remove v and all its incident edges from the graph. 3. Repeat until all the edges are covered. How good is this algorithm?
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7 Vertex Cover: Greedy Algorithm 1 OPT = 6 , all red vertices. SOL = 11 , if we are unlucky in breaking ties. First we might choose all the green vertices. Then we might choose all the blue vertices. And then we might choose all the orange vertices.
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Vertex Cover: Greedy Algorithm 1 k! vertices of degree k Generalizing the example! k!/k vertices of degree k k!/(k-1) vertices of degree k-1 k! vertices of degree 1 OPT = k!, all top vertices. SOL =
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L15-cover - Introduction to Approximation Algorithms...

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