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Lecture 21 Approximation Algorithms

Lecture 21 Approximation Algorithms. Introduction. Earlier Results on Approximations. Vertex-Cover Traveling Salesman Problem Knapsack Problem. Performance Ratio. Constant-Approximation. c-approximation is a polynomial-time approximation satisfying :

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Lecture 21 Approximation Algorithms

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  1. Lecture 21Approximation Algorithms Introduction

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

  3. Performance Ratio

  4. 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

  5. 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.

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

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

  8. Traveling Salesman with triangular inequality • Traveling around a minimum spanning tree is a 2-approximation.

  9. Traveling Salesman with Triangular Inequality • Minimum spanning tree + minimum-length perfect matching onodd verticesis 1.5-approximation

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

  11. Lower Bound 1+ε 1+ε 1 1 1+ε 1+ε 1+ε

  12. Traveling Salesman without Triangular Inequality Theorem For any constant c> 0, TSP has no c-approximation unless NP=P. Given a graph G=(V,E), define a distance table on V as follows:

  13. Contradition Argument • Suppose c-approximation exists. Then we have a polynomial-time algorithm to solve Hamiltonian Cycle as follow: C-approximation solution < cn if and only if G has a Hamiltonian cycle

  14. Knapsack

  15. 2-approximation

  16. PTAS • A problem has a PTAS (polynomial-time approximation scheme) if for any ε > 0, it has a (1+ε)-approximation.

  17. Knapsack has PTAS • Classify: for i < m, ci< a= cG, for i > m+1, ci > a. • Sort • For

  18. Proof.

  19. Time

  20. Fully PTAS • A problem has a fully PTAS if for any ε>0, it has (1+ε)-approximation running in time poly(n,1/ε).

  21. Fully FTAS for Knapsack

  22. Pseudo Polynomial-time Algorithm for Knapsak • Initially,

  23. Time • outside loop: O(n) • Inside loop: O(nM) where M=max ci • Core: O(n log (MS)) • Total O(n M log (MS)) • Since input size is O(n log (MS)), this is a pseudo-polynomial-time due to M=2 3 log M

  24. Lecture 3 Complexity of Approximation • L-reduction • Two subclass of PTAS • Set-cover ((ln n)-approximation) • Independent set (n -approximation) c

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