Unlabeled Multi-Robot Motion Planning with Tighter Separation Bounds

Authors Bahareh Banyassady , Mark de Berg , Karl Bringmann, Kevin Buchin , Henning Fernau , Dan Halperin , Irina Kostitsyna , Yoshio Okamoto , Stijn Slot

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Author Details

Bahareh Banyassady
  • Zuse Institute Berlin, Germany
Mark de Berg
  • TU Eindhoven, The Netherlands
Karl Bringmann
  • Universität des Saarlandes, Saarbrücken, Germany
  • Max Planck Institute for Informatics, Saarbrücken, Germany
Kevin Buchin
  • TU Dortmund, Germany
Henning Fernau
  • Universität Trier, Germany
Dan Halperin
  • Tel Aviv University, Israel
Irina Kostitsyna
  • TU Eindhoven, The Netherlands
Yoshio Okamoto
  • The University of Electro-Communications, Tokyo, Japan
Stijn Slot
  • Adyen, Amsterdam, The Netherlands


This research was initiated at the Lorentz-Center Workshop on Fixed-Parameter Computational Geometry, 2018. We thank Gerhard Woeginger for discussions during the workshop.

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Bahareh Banyassady, Mark de Berg, Karl Bringmann, Kevin Buchin, Henning Fernau, Dan Halperin, Irina Kostitsyna, Yoshio Okamoto, and Stijn Slot. Unlabeled Multi-Robot Motion Planning with Tighter Separation Bounds. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 12:1-12:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


We consider the unlabeled motion-planning problem of m unit-disc robots moving in a simple polygonal workspace of n edges. The goal is to find a motion plan that moves the robots to a given set of m target positions. For the unlabeled variant, it does not matter which robot reaches which target position as long as all target positions are occupied in the end. If the workspace has narrow passages such that the robots cannot fit through them, then the free configuration space, representing all possible unobstructed positions of the robots, will consist of multiple connected components. Even if in each component of the free space the number of targets matches the number of start positions, the motion-planning problem does not always have a solution when the robots and their targets are positioned very densely. In this paper, we prove tight bounds on how much separation between start and target positions is necessary to always guarantee a solution. Moreover, we describe an algorithm that always finds a solution in time O(n log n + mn + m²) if the separation bounds are met. Specifically, we prove that the following separation is sufficient: any two start positions are at least distance 4 apart, any two target positions are at least distance 4 apart, and any pair of a start and a target positions is at least distance 3 apart. We further show that when the free space consists of a single connected component, the separation between start and target positions is not necessary.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • motion planning
  • computational geometry
  • simple polygon


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