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Part V Theory of Approximate Computation

Part V Theory of Approximate Computation. Lecture 5-1 Approximation for NP-hard Problems. Earlier Results on Approximations. Vertex-Cover Traveling Salesman Problem Knapsack Problem. Performance Ratio. Constant-Approximation.

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Part V Theory of Approximate Computation

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  1. Part V Theory of Approximate Computation

  2. Lecture 5-1 Approximation for NP-hard Problems

  3. Earlier Results on Approximations • Vertex-Cover • Traveling Salesman Problem • Knapsack Problem

  4. Performance Ratio

  5. Constant-Approximation • c-approximation is apolynomial-timeapproximation satisfying: 1 < approx(input)/opt(input) < c for MIN or 1 < opt(input)/approx(input) < c for MAX

  6. Vertex Cover • Given a graph G=(V,E), find a minimum subset C of vertices such that every edge is incident to a vertex in C.

  7. Vertex-Cover • The vertex set of a maximal matching gives 2-approximation, i.e., approx / opt < 2

  8. Traveling Salesman • Given n cities with a distance table, find a minimum total-distance tour to visit each city exactly once.

  9. Special Case Theorem • Traveling around a minimum spanning tree is a 2-approximation.

  10. Theorem • Minimum spanning tree + minimum-length perfect matching on odd vertices is 1.5-approximation

  11. Minimum perfect matching on odd vertices has weight at most 0.5 opt.

  12. Knapsack

  13. Theorem Proof.

  14. Theorem

  15. Algorithm • Classify: for i < m, ci< a= cG, for i > m+1, ci > a. • Sort • For Why?

  16. Why?

  17. Proof.

  18. Time

  19. Thanks, end.

  20. Theorem Proof: Given a graph G=(V,E), define a distance table on V as follows:

  21. Contradiction Argument • Suppose r-approximation exists. Then we have a polynomial-time algorithm to solve Hamiltonian Cycle as follow: r-approximation solution <r |V| if and only if G has a Hamiltonian cycle

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