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Polynomial-Time Algorithms for the Touring Rays and Related Problems

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Algorithmic Aspects in Information and Management (AAIM 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12290))

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Abstract

The touring rays problem, which is also known as the traveling salesman problem for rays in the plane, asks to compute the shortest (closed) route that tours or intersects n given rays. We show that it can be reduced to the problem of computing a shortest route that intersects a set of ray-segments, inside a circle; at least one endpoint of every ray-segment is on the circle. Moreover, computing the shortest route intersecting all ray-segments in the circle is related to the solution of the well-known watchman route problem. Our method is further extended to solve the minimum-perimeter intersecting polygon problem, which asks for a (convex) polygon P of minimum perimeter such that P contains at least one point of every given line segment. Both of our algorithms run in \(O(n^5)\) time, and they solve two long-standing open problems in computational geometry.

This work was partially supported by JSPS KAKENHI Grant Number 15K00023 and 20K11683.

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Notes

  1. 1.

    The circle \(\mathcal C\) in Figs. 2, 4 and 5 is not drawn exactly, as it only shows an efficiently large region that contains \(OPT_s(\mathcal{S})\).

  2. 2.

    Although the order of pockets is defined, no order of ray-segments is specified.

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Authors and Affiliations

  1. Tokai University, 4-1-1 Kitakaname, Hiratsuka, 259-1292, Japan

    Xuehou Tan

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  1. Xuehou Tan
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Correspondence to Xuehou Tan .

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Editors and Affiliations

  1. Zhejiang Normal University, Jinhua, China

    Zhao Zhang

  2. Texas Southern University, Houston, TX, USA

    Wei Li

  3. University of Texas at Dallas, Richardson, TX, USA

    Ding-Zhu Du

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Tan, X. (2020). Polynomial-Time Algorithms for the Touring Rays and Related Problems. In: Zhang, Z., Li, W., Du, DZ. (eds) Algorithmic Aspects in Information and Management. AAIM 2020. Lecture Notes in Computer Science(), vol 12290. Springer, Cham. https://doi.org/10.1007/978-3-030-57602-8_1

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  • DOI: https://doi.org/10.1007/978-3-030-57602-8_1

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-57601-1

  • Online ISBN: 978-3-030-57602-8

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