Part V Theory of Approximate Computation

1. Part V Theory of Approximate Computation

2. Lecture 25 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

16. 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