[Paper Review] Coordinating the Motion of Labeled Discs with Optimality Guarantees under Extreme Density

Rupesh Chinta, Shuai D. Han+1 more|arXiv (Cornell University)|Jul 9, 2018

Robotic Path Planning Algorithms14 references3 citations

TL;DR

This paper presents a polynomial-time algorithm that guarantees constant-factor time-optimality for routing labeled unit discs in 2D environments with over 50% robot density—achieving this by transforming the continuous motion problem into a discrete routing task on a triangular grid. A complementary fast ILP-based method further improves solution quality, enabling near-optimal plans even under extreme packing conditions.

ABSTRACT

We push the limit in planning collision-free motions for routing uniform labeled discs in two dimensions. First, from a theoretical perspective, we show that the constant-factor time-optimal routing of labeled discs can be achieved using a polynomial-time algorithm with robot density over $50\%$ in the limit (i.e., over half of the workspace may be occupied by the discs). Second, from a more practical standpoint, we provide a high performance algorithm that computes near-optimal (e.g., $1.x$) solutions under the same density setting.

Motivation & Objective

• Address the challenge of time-optimal motion planning for labeled rigid bodies (unit discs) in extremely dense 2D environments.
• Overcome the computational intractability of dense multi-robot routing by enabling high robot density (over 50%) while preserving optimality guarantees.
• Develop a complete, deterministic, and polynomial-time algorithm for time-optimal routing under extreme density, surpassing prior methods limited to ~23% density.
• Provide a practical, high-performance ILP-based algorithm for near-optimal solutions in dense configurations without polynomial time guarantees.
• Enable scalable and provably efficient routing in applications such as automated warehouses, where high spatial and energy efficiency are critical.

Proposed method

• Transform the continuous motion problem of labeled unit discs into a discrete multi-robot routing problem on a triangular grid, enabled by a minimum center-to-center separation of 8/3.
• Leverage a grid-based transformation that ensures minimal optimality loss and allows the use of discrete routing algorithms with theoretical guarantees.
• Design a low-polynomial-time algorithm for routing on the triangular grid that enforces the constraint that no two discs occupy the same triangle simultaneously.
• Adapt a fast ILP-based method from prior work to the triangular grid, incorporating constraints for concurrent movement avoidance on shared triangles.
• Introduce a novel discretization procedure, DiscretizeOLDR, which maps continuous configurations to the triangular grid while preserving separation and enabling optimality analysis.
• Apply k-way split heuristics to scale the ILP-based solver to larger problem instances, improving runtime and solution quality on large-scale dense environments.

Experimental results

Research questions

• RQ1 Can time-optimal routing of labeled unit discs be achieved under robot densities exceeding 50% of the workspace, with provable optimality guarantees?
• RQ2 How can the continuous motion planning problem for rigid bodies be effectively transformed into a discrete routing problem on a triangular grid while preserving optimality?
• RQ3 What is the impact of discretization on solution quality, particularly in high-density configurations approaching theoretical limits?
• RQ4 Can a practical, fast ILP-based algorithm outperform existing methods in terms of solution quality and scalability under extreme density?
• RQ5 How does the triangular grid compare to hexagonal grids in terms of achievable density, solution quality, and computational efficiency for multi-robot routing?

Key findings

• The proposed polynomial-time algorithm guarantees O(1) time-optimality for labeled disc routing with robot density exceeding 50%, achieved via a triangular grid discretization with 8/3 center separation.
• The ILP-based method (TriILP) computes near-optimal solutions, solving 110-robot instances in 55 seconds with a 1.65-optimality ratio using 4-way split heuristics.
• For a 232-vertex triangular grid, TriILP solved 140-robot problems in 55 seconds with 8-way split, achieving a 1.65-optimality ratio.
• The method achieved a 1.5-optimal solution in 2.1 seconds for a 20-robot compact instance with 8/3 separation, demonstrating high efficiency on tight configurations.
• With 8-way split, the full pipeline (DiscretizeOLDR + TriILP) solved 120-robot problems in 21.93 seconds with a 3.88 optimality ratio, showing scalability and robustness.
• Performance comparison showed that TriILP outperforms HexILP in both solution quality and scalability, despite higher model size, due to better density utilization and lower discretization loss.