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Bitonicity of Euclidean TSP in Narrow Strips EuroCG presentation Henk Alkema, Mark de Berg, and Sndor Kisfaludi-Bak Department of Mathematics and Computer Science Introduction The Euclidean Travelling Salesman Problem In red: added

  1. Bitonicity of Euclidean TSP in Narrow Strips EuroCG presentation Henk Alkema, Mark de Berg, and SΓ‘ndor Kisfaludi-Bak Department of Mathematics and Computer Science

  2. Introduction

  3. The Euclidean Travelling Salesman Problem In red: added commentary to make the slides readable

  4. The Euclidean Travelling Salesman Problem Find a shortest tour visiting all points

  5. Euclidean TSP in narrow strips Find a shortest tour visiting all points Likely to be a bitonic tour A tour is bitonic if it crosses any vertical line at most twice

  6. Motivation 𝑒 -dimensional Euclidean TSP: NP-hard Can be solved in 2 𝑃 π‘œ 1βˆ’1/𝑒 time ETH-tight π‘œ for 𝑒 = 2 2 𝑃 Bitonic tours: 𝑃(π‘œ log 2 π‘œ)

  7. Problem description 𝑄 = {π‘ž 1 , π‘ž 2 , … , π‘ž π‘œ } point set in 0, π‘œ Γ— 0, πœ€ 𝑦 -coordinate of π‘ž 𝑗 is exactly 𝑗

  8. Bitonicity of Euclidean TSP in Narrow Strips

  9. Bitonicity of Euclidean TSP in Narrow Strips Theorem 1. If πœ€ ≀ 2 2 , there exists a shortest tour that is bitonic. This bound is tight. Construction for πœ€ > 2 2 :

  10. Bitonicity of Euclidean TSP in Narrow Strips Theorem 1. If πœ€ ≀ 2 2 , there exists a shortest tour that is bitonic. This bound is tight. Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

  11. Bitonicity of Euclidean TSP in Narrow Strips Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

  12. Bitonicity of Euclidean TSP in Narrow Strips Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

  13. Bitonicity of Euclidean TSP in Narrow Strips Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

  14. Bitonicity of Euclidean TSP in Narrow Strips Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

  15. Bitonicity of Euclidean TSP in Narrow Strips Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

  16. Bitonicity of Euclidean TSP in Narrow Strips Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

  17. Bitonicity of Euclidean TSP in Narrow Strips Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

  18. Bitonicity of Euclidean TSP in Narrow Strips Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

  19. Bitonicity of Euclidean TSP in Narrow Strips Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

  20. Bitonicity of Euclidean TSP in Narrow Strips Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

  21. Bitonicity of Euclidean TSP in Narrow Strips Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

  22. Bitonicity of Euclidean TSP in Narrow Strips Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

  23. Bitonicity of Euclidean TSP in Narrow Strips Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€² An edge set E is superior to an edge set F if - The sum of the lengths of the edges of E is strictly less than that of F, or - The sums are equal, but - No vertical line crosses E strictly more times than F, and - There exists a vertical line which crosses E strictly fewer times than F

  24. Bitonicity of Euclidean TSP in Narrow Strips Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€² Step 1: Superior edge set exists if β€˜interesting’ points have consecutive 𝑦 -coordinates β‡’ A superior edge set always exists β€˜Interesting’ points are those which cross the vertical line we are currently looking at during our sweep from right to left

  25. Bitonicity of Euclidean TSP in Narrow Strips Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€² Step 1: Superior edge set exists if β€˜interesting’ points have consecutive 𝑦 -coordinates β‡’ A superior edge set always exists Step 2: Superior edge set exists if β€˜interesting’ points have consecutive 𝑦 -coordinates

  26. Bitonicity of Euclidean TSP in Narrow Strips Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

  27. Bitonicity of Euclidean TSP in Narrow Strips Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€² The exact connections are unimportant, but their connections are; the new set of edges must still form a tour together with the black edges

  28. Bitonicity of Euclidean TSP in Narrow Strips Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆ β€² In blue, a alternative set of edges. Note that moving points along the red edges can only make blue β€˜less’ superior

  29. Bitonicity of Euclidean TSP in Narrow Strips Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

  30. Bitonicity of Euclidean TSP in Narrow Strips Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

  31. Bitonicity of Euclidean TSP in Narrow Strips Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

  32. Bitonicity of Euclidean TSP in Narrow Strips Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€² If you whish to move a vertex with two red edges connected, things are slightly more complicated…

  33. Bitonicity of Euclidean TSP in Narrow Strips Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€² Split the vertex into two, adding a connection between them (they are still on the same spot, but slightly displaced in the figure for clarity)

  34. Bitonicity of Euclidean TSP in Narrow Strips Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€² Then, you can move one of them as normal

  35. Bitonicity of Euclidean TSP in Narrow Strips Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

  36. Bitonicity of Euclidean TSP in Narrow Strips Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€² Step 1: Superior edge set exists if β€˜interesting’ points have consecutive 𝑦 -coordinates β‡’ A superior edge set always exists Step 2: Superior edge set exists if β€˜interesting’ points have consecutive 𝑦 -coordinates

  37. Bitonicity of Euclidean TSP in Narrow Strips Step 2: Superior edge set exists if β€˜interesting’ points have consecutive 𝑦 -coordinates Proof sketch: Case distinction on the connections between the β€˜interesting’ points

  38. Bitonicity of Euclidean TSP in Narrow Strips All six possible cases. Points in grey blocks can have any horizontal ordering

  39. Bitonicity of Euclidean TSP in Narrow Strips Proof sketch: Case distinction on the connections between the β€˜interesting’ points For each case, this can be proven either algebraically, or by computer assistance The figure to the right gives the bound (both horizontal orderings of points 4 and 5 give the same bound of 2√2 )

  40. Questions? Feel free to let us know!

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