Abstract
Motivated by the question of optimal facility placement, the classical p-dispersion problem seeks to place a fixed number of equally sized non-overlapping circles of maximal possible radius into a subset of the plane. While exact solutions to this problem may be found for placement into particular sets, the problem is provably NP-complete for general sets, and existing work is largely restricted to geometrically simple sets. This paper makes two contributions to the theory of p-dispersion. First, we propose a computationally feasible suboptimal approach to the p-dispersion problem for all non-convex polygons. The proposed method, motivated by the mechanics of the p-body problem, considers circle centers as continuously moving objects in the plane and assigns repulsive forces between different circles, as well as circles and polygon boundaries, with magnitudes inversely proportional to the corresponding distances. Additionally, following the motivating application of optimal facility placement, we consider existence of additional hard upper or lower distance bounds on pairs of circle centers, and adapt the proposed method to provide a p-dispersion solution that provably respects such constraints. We validate our proposed method by comparing it with previous exact and approximate methods for p-dispersion. The method quickly produces near-optimal results for a number of containers.
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There is no data or material associated with the work. Numerical examples of the work can largely be reproduced using the code at https://github.com/zd6/RApDoP.
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Funding
This work was supported by NASA’s Space Technology Research Grants program for Early Stage Innovations under the grant “Safety-Constrained and Efficient Learning for Resilient Autonomous Space Systems.”
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A software implementation of the proposed algorithm is available at https://github.com/zd6/RApDoP.
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Dai, Z., Xu, K. & Ornik, M. Repulsion-based p-dispersion with distance constraints in non-convex polygons. Ann Oper Res 307, 75–91 (2021). https://doi.org/10.1007/s10479-021-04281-z
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DOI: https://doi.org/10.1007/s10479-021-04281-z