Sub-1.5 Time-Optimal Multi-Robot Path Planning on Grids in Polynomial Time (2201.08976v3)

Abstract: Graph-based multi-robot path planning (MRPP) is NP-hard to optimally solve. In this work, we propose the first low polynomial-time algorithm for MRPP achieving 1--1.5 asymptotic optimality guarantees on makespan for random instances under very high robot density, with high probability. The dual guarantee on computational efficiency and solution optimality suggests our proposed general method is promising in significantly scaling up multi-robot applications for logistics, e.g., at large robotic warehouses. Specifically, on an $m_1\times m_2$ gird, $m_1 \ge m_2$, our RTH (Rubik Table with Highways) algorithm computes solutions for routing up to $\frac{m_1m_2}{3}$ robots with uniformly randomly distributed start and goal configurations with a makespan of $m_1 + 2m_2 + o(m_1)$, with high probability. Because the minimum makespan for such instances is $m_1 + m_2 - o(m_1)$, also with high probability, RTH guarantees $\frac{m_1+2m_2}{m_1+m_2}$ optimality as $m_1 \to \infty$ for random instances with up to $\frac{1}{3}$ robot density, with high probability. $\frac{m_1+2m_2}{m_1+m_2} \in (1, 1.5]$. Alongside this key result, we also establish a series of related results supporting even higher robot densities and environments with regularly distributed obstacles, which directly map to real-world parcel sorting scenarios. Building on the baseline methods with provable guarantees, we have developed effective, principled heuristics that further improve the computed optimality of the RTH algorithms. In extensive numerical evaluations, RTH and its variants demonstrate exceptional scalability as compared with methods including ECBS and DDM, scaling to over $450 \times 300$ grids with $45,000$ robots, and consistently achieves makespan around $1.5$ optimal or better, as predicted by our theoretical analysis.

• The paper introduces the RTH algorithm, a polynomial-time method leveraging Rubik Tables and highway heuristics for sub-1.5 time-optimal multi-robot path planning.
• RTH achieves sub-1.5 optimality for high robot densities (up to 1/3 capacity) and efficiently solves instances with up to 45,000 robots on large grids.
• This work scales up multi-robot path planning for logistics like automated warehouses and provides theoretical groundwork for improved efficiency and real-time applications.

Overview of Sub-1.5 Time-Optimal Multi-Robot Path Planning on Grids in Polynomial Time

This paper introduces an innovative algorithm designed for multi-robot path planning (MRPP) on grid-based environments. The problem addressed is known for its computational complexity and is generally regarded as NP-hard, even under simplified grid configurations. The proposed algorithm, referred to as RTH (Rubik Table with Highways), achieves time-optimal solutions within a $1$ to $1.5$ factor of the best possible makespan for random instances with high robot densities, computed in polynomial time.

Main Contributions

1. Algorithmic Methodology: The RTH algorithm leverages the concept of Rubik Tables—a global rearrangement method—enhanced by highway heuristics to navigate high densities of robots on grids. This is effectively coupled with a series of matching techniques that facilitate a highly coordinated yet computationally feasible path finding process.
2. Optimality and Scalability: The authors provide evidence that RTH reaches sub-1.5 asymptotic optimality with high probability while handling high robot densities, up to $\frac{1}{3}$ of the total grid capacity. In practical settings, the algorithm demonstrates the ability to efficiently solve instances with up to $45,000$ robots on large $450 \times 300$ grids, maintaining an optimality ratio around $1.5$ or better.
3. Algorithmic Enhancements: Beyond the foundational RTH method, additional improvements such as RTH-IP and RTH-LBA are introduced. These variants incorporate integer programming-based and linear bottleneck assignment heuristics respectively, further optimizing the matchings and thereby improving makespan efficiency.
4. Handling Obstacles and Higher Densities: The authors also adapt their solution to accommodate scenarios with regularly distributed obstacles and provide an extended algorithm, RTLM, which supports densities reaching $\frac{1}{2}$ with comparable theoretical guarantees.

Numerical Results

The paper substantiates its claims through extensive computational experiments:

• Optimality Ratios: For random configurations up to full capacity, the RTM variant achieves makespan optimality ranging from $7$ to $10.5+$, while RTH and RTLM deliver sub-$1.5$ results in numerous tested scenarios and grid sizes.
• Computation Time: Despite the intricate coordination needed for high-density problem spaces, RTH and its derivatives outperform contemporary solvers like ECBS and DDM in terms of scalability and speed, particularly in large settings.

Implications and Future Directions

This work significantly scales up the potential applications of MRPP in logistics domains, particularly in settings such as automated warehouse systems and parcel sorting. The ability to handle complex environments with computational guarantees allows for practical deployment in real-time operations where density and efficiency are critical.

From a theoretical perspective, this paper opens avenues for further exploration into more efficient line shuffle routines, improved matchings for better solution optimality at varied densities, and supporting additional complex robot models. It also lays the groundwork for addressing life-long MRPP scenarios, where maintaining high throughput in real-time applications could yield significant advancements.

In conclusion, the algorithmic contributions by Guo and Yu address a pivotal aspect of multi-agent planning by proposing a tractable solution that balances computational feasibility with optimality, providing a substantial improvement over existing approaches in both theoretical guarantees and practical scalability.