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The touring rays and related problems
Theoretical Computer Science ( IF 1.1 ) Pub Date : 2021-03-17 , DOI: 10.1016/j.tcs.2021.03.005
Xuehou Tan

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 for a given set of line segments, P contains at least one point of every line segment. Both of our algorithms run in O(n5) time, and they solve two long-standing open problems in computational geometry.



中文翻译:

旅游射线及相关问题

旅行射线问题(也称为飞机上射线的旅行推销员问题)要求计算游览或相交n个给定射线的最短(闭合)路线。我们证明了它可以简化为计算与圆内的一组射线段相交的最短路径的问题。每个射线段的至少一个端点在圆上。此外,计算与圆中所有射线段相交的最短路径与解决众所周知的守望者路径问题有关。我们的方法进一步扩展为解决最小周长相交多边形问题,该问题要求最小周长的(凸)多边形P使得对于给定的一组线段P每个线段至少包含一个点。我们两种算法都在Øñ5 时间,它们解决了计算几何学中两个长期存在的开放性问题。

更新日期:2021-04-08
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