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Geometric Multicut: Shortest Fences for Separating Groups of Objects in the Plane

Abrahamsen, M., Giannopoulos, P. ORCID: 0000-0002-6261-1961, Loffler, M. and Rote, G. (2020). Geometric Multicut: Shortest Fences for Separating Groups of Objects in the Plane. Discrete & Computational Geometry, doi: 10.1007/s00454-020-00232-w

Abstract

We study the following separation problem: Given a collection of pairwise disjoint coloured objects in the plane with k different colours, compute a shortest “fence” F, i.e., a union of curves of minimum total length, that separates every pair of objects of different colours. Two objects are separated if F contains a simple closed curve that has one object in the interior and the other in the exterior. We refer to the problem as GEOMETRIC k-CUT, as it is a geometric analog to the well-studied multicut problem on graphs. We first give an O(n4log3n)-time algorithm that computes an optimal fence for the case where the input consists of polygons of two colours with n corners in total. We then show that the problem is NP-hard for the case of three colours. Finally, we give a randomised 4/3⋅1.2965-approximation algorithm for polygons and any number of colours.

Publication Type: Article
Additional Information: This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Subjects: Q Science > QA Mathematics
Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Departments: School of Mathematics, Computer Science & Engineering > Computer Science > giCentre
Date Deposited: 14 Aug 2020 11:11
URI: https://openaccess.city.ac.uk/id/eprint/24746
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