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DOI: 10.4230/LIPIcs.SoCG.2025.5

Optimal Motion Planning for Two Square Robots in a Rectilinear Environment

Authors: Pankaj K. Agarwal, Mark de Berg, Benjamin Holmgren, Alex Steiger, and Martijn Struijs

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract

Let W ⊂ ℝ² be a rectilinear polygonal environment (that is, a rectilinear polygon potentially with holes) with a total of n vertices, and let A,B be two robots, each modeled as an axis-aligned unit square, that can move rectilinearly inside W. The goal is to compute an optimal collision-free motion plan π for A and B between a given pair of source and target configurations. We study two variants of this problem and obtain the following results. - Min-Sum: Here the goal is to compute a motion plan that minimizes the sum of the lengths of the paths of the robots. We present an O(n⁴log n)-time algorithm for computing an optimal solution to the min-sum problem. This is the first polynomial-time algorithm to compute an optimal, collision-free motion of two robots amid obstacles in a planar polygonal environment. - Min-Makespan: Here the robots can move with at most unit speed, and the goal is to compute a motion plan that minimizes the maximum time taken by a robot to reach its target location. We prove that the min-makespan variant is NP-hard.

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Pankaj K. Agarwal, Mark de Berg, Benjamin Holmgren, Alex Steiger, and Martijn Struijs. Optimal Motion Planning for Two Square Robots in a Rectilinear Environment. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 5:1-5:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{agarwal_et_al:LIPIcs.SoCG.2025.5,
  author =	{Agarwal, Pankaj K. and de Berg, Mark and Holmgren, Benjamin and Steiger, Alex and Struijs, Martijn},
  title =	{{Optimal Motion Planning for Two Square Robots in a Rectilinear Environment}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{5:1--5:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.5},
  URN =		{urn:nbn:de:0030-drops-231577},
  doi =		{10.4230/LIPIcs.SoCG.2025.5},
  annote =	{Keywords: Computational geometry, motion planning, multiple robots, rectilinear paths}
}

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Optimal Motion Planning for Two Square Robots in a Rectilinear Environment

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