## Abstract

Let W ⊂ R^{2} be a planar polygonal environment (i.e., a 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 translate inside W. Given source and target placements sA, tA, sB, tB ∈ W of A and B, respectively, the goal is to compute a collision-free motion plan π^{∗}, i.e., a motion plan that continuously moves A from sA to tA and B from sB to tB so that A and B remain inside W and do not collide with each other during the motion. Furthermore, if such a plan exists, then we wish to return a plan that minimizes the sum of the lengths of the paths traversed by the robots. Given W, sA, tA, sB, tB and a parameter ε > 0, we present an (Equation presented)-approximation algorithm for this problem. We are not aware of any polynomial-time algorithm for this problem, nor do we know whether the problem is NP-Hard. Our result is the first polynomial-time (1 + ε)-approximation algorithm for an optimal motion-planning problem involving two robots moving in a polygonal environment.

Original language | English |
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Pages | 4942-4962 |

Number of pages | 21 |

DOIs | |

State | Published - 2024 |

Event | 35th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2024 - Alexandria, United States Duration: 7 Jan 2024 → 10 Jan 2024 |

### Conference

Conference | 35th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2024 |
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Country/Territory | United States |

City | Alexandria |

Period | 7/01/24 → 10/01/24 |