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arXiv · 1703.05549

Minimum Perimeter-Sum Partitions in the Plane

Mikkel Abrahamsen·Mark de Berg·Kevin Buchin·Mehran Mehr·Ali D. Mehrabi
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Abstract

Let $P$ be a set of $n$ points in the plane. We consider the problem of partitioning $P$ into two subsets $P_1$ and $P_2$ such that the sum of the perimeters of $\text{CH}(P_1)$ and $\text{CH}(P_2)$ is minimized, where $\text{CH}(P_i)$ denotes the convex hull of $P_i$. The problem was first studied by Mitchell and Wynters in 1991 who gave an $O(n^2)$ time algorithm. Despite considerable progress on related problems, no subquadratic time algorithm for this problem was found so far. We present an exact algorithm solving the problem in $O(n \log^2 n)$ time and a $(1+\varepsilon)$-approximation algorithm running in $O(n + 1/\varepsilon^2\cdot\log^2(1/\varepsilon))$ time.

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Mikkel Abrahamsen, Mark de Berg, Kevin Buchin, Mehran Mehr, Ali D. Mehrabi. 2017-03-16. Minimum Perimeter-Sum Partitions in the Plane. https://doi.org/10.1007/s00454-019-00059-0

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