Optimal Motion Planning for Two Square Robots in a Rectilinear Environment

2026-01-27

International Symposium on Computational Geometry

AI Summary

This study addresses the problem of collision-free motion planning for two axis-aligned unit-square robots in a rectilinear polygonal environment with obstacles, focusing on two optimization objectives: minimizing the total path length (Min-Sum) and minimizing the completion time (Min-Makespan). For the Min-Sum objective, the authors present the first polynomial-time exact algorithm with a time complexity of $O(n^4 \log n)$. In contrast, they prove that the Min-Makespan problem is NP-hard. This work not only enables efficient optimal solutions for the Min-Sum variant but also establishes a fundamental computational complexity lower bound for the Min-Makespan variant, thereby advancing the theoretical understanding of multi-robot motion planning in constrained geometric environments.

Technology Category

• Intelligent Robots: Motion and Path Planning
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Security and privacy of machine learning and AI applications
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Let $\mathcal{W} \subset \mathbb{R}^2$ 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 $\mathcal{W}$. The goal is to compute a collision-free motion plan $\boldsymbol{\pi}$, that is, a motion plan that continuously moves $A$ from $s_A$ to $t_A$ and $B$ from $s_B$ to $t_B$ so that $A$ and $B$ remain inside $\mathcal{W}$ and do not collide with each other during the motion. We study two variants of this problem which are focused additionally on the optimality of $\boldsymbol{\pi}$, and obtain the following results. 1. 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 $O(n^4\log{n})$-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. 2. 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.

Problem

• motion planning
• collision-free
• rectilinear environment
• multi-robot
• optimality

Innovation

• motion planning
• collision-free
• rectilinear environment
• min-sum optimization
• NP-hardness