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Optimal Motion Planning for Two Square Robots in a Rectilinear Environment

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

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ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Computational geometry
  • motion planning
  • multiple robots
  • rectilinear paths

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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.

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.SoCG.2025.5

Author Details

Pankaj K. Agarwal
  • Department of Computer Science, Duke University, Durham, NC, USA
Mark de Berg
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Benjamin Holmgren
  • Department of Computer Science, Duke University, Durham, NC, USA
Alex Steiger
  • Department of Computer Science, Duke University, Durham, NC, USA
Martijn Struijs
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands

Funding

  • Agarwal, Pankaj K.: Supported by NSF grants CCF-20-07556, CCF-22-23870, and IIS-24-02823, and by the Binational Science Foundation Grant 2022131.
  • de Berg, Mark: Supported by the Dutch Research Council (NWO) through Gravitation-grant NETWORKS-024.002.003.
  • Holmgren, Benjamin: Supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Department of Energy Computational Science Graduate Fellowship under Award Number DE-SC0024386.

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