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Algorithms Design and Analysis: PTAS for Euclidean TSP

Algorithms Design and Analysis: PTAS for Euclidean TSP. Prof. Dr. Jinxing Xie Dept. of Mathematical Sciences Tsinghua University, Beijing 100084, China Email:jxie@math.tsinghua.edu.cn http://faculty.math.tsinghua.edu.cn/~jxie Voice:(86-10)62787812 Fax:(86-10)62785847. Reference.

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Algorithms Design and Analysis: PTAS for Euclidean TSP

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  1. Algorithms Design and Analysis:PTAS for Euclidean TSP Prof. Dr. Jinxing Xie Dept. of Mathematical Sciences Tsinghua University, Beijing 100084, China Email:jxie@math.tsinghua.edu.cn http://faculty.math.tsinghua.edu.cn/~jxie Voice:(86-10)62787812 Fax:(86-10)62785847

  2. Reference S. Arora, Polynomial time Approximation Schemes for Euclidean TSP and other Geometric Problems. Proc of the 37th Annual Symposium on the Foundations of Computer Science, 2-11, 1996. • [Note] • Euclidean TSP is a special case of TSP with triangle inequality • We always mean the Euclidean TSP in two dimensions

  3. Simplifying assumptions about the locations of the cities Given any finite set S of points (cities) in the plane, the bounding box of S is the smallest rectangle with sides parallel to the coordinate axes which contains all of S. The size of a rectangle is the length of its longest side. [Claim] Suppose there is an approximation scheme for instances of the Euclidean TSP in which the n cities lie in a bounding box of size n2 and the distance between any two cities is at least 1, or 0. Then there is an approximation scheme for all instances of the Euclidean TSP.

  4. Line-partition & 1/3:2/3-tiling 2

  5. How deep can the recursion? • Note that: • The sizes of the rectangles need not decrease at each stage because the first partition of a square produces two rectangles of the same size as the original square. • Also, if the cities are all near the sides of the rectangle, no valid line-partition will cut the longest side s into pieces of length at least |s|/3.

  6. How deep can the recursion? • However, • after at most four levels of recursion, all the remaining rectangles will either have size at most 2|s|/3 or will have no cities in their interior.

  7. Portals & m-light

  8. Example We allow a tour which goes from city a to city b and from city c to city d via portal p, this is an example of a tour which meets itself at a portal. However we do not allow a tour which goes from city a to city d and from city b to city c both via portal p, this is a self-crossing.

  9. Theorem

  10. Deal with crossing

  11. Deal with crossing

  12. Proof of the 2nd Part of the Theorem

  13. Dynamic programming algorithm

  14. Completing the proof

  15. How do you think about the proof? • it is difficult • it is interesting Thus it is challenging! However, is there an FPTAS for the problem?

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