1 / 22

Approximation Theory Lecture: NP-Hard Problems & Performance Ratios

Explore approximations for NP-hard problems including Vertex Cover, Traveling Salesman, and Knapsack Problem. Learn about performance ratios, constant-approximations, and algorithms for optimization. Understand the theory and applications in computational complexity.

spinella
Download Presentation

Approximation Theory Lecture: NP-Hard Problems & Performance Ratios

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related