Bitonicity of Euclidean TSP in Narrow Strips EuroCG presentation - - PowerPoint PPT Presentation

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Bitonicity of Euclidean TSP in Narrow Strips EuroCG presentation - - PowerPoint PPT Presentation

Jan 09, 2023 246 likes β€’658 views

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

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Department of Mathematics and Computer Science Henk Alkema, Mark de Berg, and SΓ‘ndor Kisfaludi-Bak

EuroCG presentation

Bitonicity of Euclidean TSP in Narrow Strips

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Introduction

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The Euclidean Travelling Salesman Problem

In red: added commentary to make the slides readable

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The Euclidean Travelling Salesman Problem

Find a shortest tour visiting all points

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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

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Motivation

𝑒-dimensional Euclidean TSP: NP-hard Can be solved in 2𝑃 π‘œ1βˆ’1/𝑒 time ETH-tight 2𝑃

π‘œ for 𝑒 = 2

Bitonic tours: 𝑃(π‘œ log2 π‘œ)

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Problem description

𝑄 = {π‘ž1, π‘ž2, … , π‘žπ‘œ} point set in 0, π‘œ Γ— 0, πœ€ 𝑦-coordinate of π‘žπ‘— is exactly 𝑗

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Bitonicity of Euclidean TSP in Narrow Strips

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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:

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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

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Bitonicity of Euclidean TSP in Narrow Strips

Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

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Bitonicity of Euclidean TSP in Narrow Strips

Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

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Bitonicity of Euclidean TSP in Narrow Strips

Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

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Bitonicity of Euclidean TSP in Narrow Strips

Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

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Bitonicity of Euclidean TSP in Narrow Strips

Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

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Bitonicity of Euclidean TSP in Narrow Strips

Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

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Bitonicity of Euclidean TSP in Narrow Strips

Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

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Bitonicity of Euclidean TSP in Narrow Strips

Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

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Bitonicity of Euclidean TSP in Narrow Strips

Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

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Bitonicity of Euclidean TSP in Narrow Strips

Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

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Bitonicity of Euclidean TSP in Narrow Strips

Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

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Bitonicity of Euclidean TSP in Narrow Strips

Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

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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
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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

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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

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Bitonicity of Euclidean TSP in Narrow Strips

Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

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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

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Bitonicity of Euclidean TSP in Narrow Strips

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

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Bitonicity of Euclidean TSP in Narrow Strips

Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

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Bitonicity of Euclidean TSP in Narrow Strips

Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

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Bitonicity of Euclidean TSP in Narrow Strips

Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

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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…

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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)

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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

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Bitonicity of Euclidean TSP in Narrow Strips

Proof sketch: Transform tour π‘ˆ into (shorter) bitonic tour π‘ˆβ€²

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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

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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

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Bitonicity of Euclidean TSP in Narrow Strips

All six possible cases. Points in grey blocks can have any horizontal ordering

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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)

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Questions?

Feel free to let us know!