[Paper Review] Constant Factor Time Optimal Multi-Robot Routing on High-Dimensional Grids in Mostly Sub-Quadratic Time

Jingjin Yu|arXiv (Cornell University)|Jan 31, 2018

Optimization and Search Problems37 references3 citations

TL;DR

This paper presents i SaG and PartitionAndFlow (PaF), two algorithms for time-optimal multi-robot routing on high-dimensional grids with maximal robot density. i SaG achieves average-case constant-factor time optimality in mostly sub-quadratic time, while PaF guarantees worst-case constant-factor time optimality with $O(d_g)$ makespan and $o(|V|^2)$ running time for arbitrary fixed dimensions $k \geq 2$, leveraging advanced matching and flow decomposition techniques.

ABSTRACT

Let $G = (V, E)$ be an $m_1 imes \ldots imes m_k$ grid. Assuming that each $v \in V$ is occupied by a robot and a robot may move to a neighboring vertex in a step via synchronized rotations along cycles of $G$, we first establish that the arbitrary reconfiguration of labeled robots on $G$ can be performed in $O(k\sum_i m_i)$ makespan and requires $O(|V|^2)$ running time in the worst case and $o(|V|^2)$ when $G$ is non-degenerate (in the current context, a grid is degenerate if it is nearly one dimensional). The resulting algorithm, iSAG, provides average case $O(1)$-approximate (i.e., constant-factor) time optimality guarantee. When all dimensions are of similar size $O(|V|^{\frac{1}{k}})$, the running time of iSAG approaches a linear $O(|V|)$. Define $d_g(p)$ as the largest distance between individual initial and goal configurations over all robots for a given problem instance $p$, building on iSAG, we develop the PartitionAndFlow (PAF) algorithm that computes $O(d_g(p))$ makespan solutions for arbitrary fixed $k \ge 2$, using mostly $o(|V|^2)$ running time. PAF provides worst case $O(1)$-approximation regarding solution time optimality. We note that the worst case running time for the problem is $Ω(|V|^2)$.

Motivation & Objective

• To develop a time-optimal multi-robot routing algorithm for high-dimensional grids with maximal robot density.
• To achieve sub-quadratic running time while maintaining constant-factor time optimality in the average case.
• To extend the solution to worst-case time-optimality guarantees using a novel flow-based routing strategy.
• To provide efficient algorithms that scale well with grid dimensionality and robot displacement.
• To enable practical deployment in dense robotic systems such as automated warehouses and container ports.

Proposed method

• i SaG uses a shuffling procedure based on bipartite matching to redistribute unlabeled robots across the grid, enabling efficient reconfiguration.
• The algorithm decomposes robot flows into unit circulations using a matching-based method, allowing for efficient flow routing.
• PaF leverages i SaG as a subroutine and applies recursive peeling and stacking of flow shapes across orthogonal dimensions to reshape flows.
• The method uses a $[6d_g]^k$-sized auxiliary grid to safely reshape and route flows between faces of the main grid.
• A key component is the use of $\Omega(d_g^{k-1})$ vertex-disjoint paths in a $\Theta(d_g)^k$ grid, enabling high-throughput flow routing.
• PaF computes a global routing plan using max-flow algorithms, with time complexity dominated by $O(d_g^k |V|)$.

Experimental results

Research questions

• RQ1 Can time-optimal multi-robot routing be achieved in mostly sub-quadratic time on high-dimensional grids?
• RQ2 Is it possible to maintain constant-factor time optimality in the worst case while reducing running time below $O(|V|^2)$?
• RQ3 How can flow reshaping be efficiently performed across multiple dimensions to enable high-throughput routing?
• RQ4 What is the role of vertex-disjoint path structures in enabling efficient routing on high-dimensional grids?
• RQ5 Can the algorithmic framework be generalized to other grid-like or continuous robotic settings?

Key findings

• i SaG achieves average-case $O(1)$-approximate time optimality with $o(|V|^2)$ running time for non-degenerate grids and $O(|V|^2)$ in the worst case.
• When all dimensions are of size $O(|V|^{1/k})$, the running time of i SaG approaches linear $O(|V|)$.
• PaF computes solutions with $O(d_g(p))$ makespan for any fixed $k \geq 2$, ensuring worst-case $O(1)$-approximation in time optimality.
• PaF runs in $O(d_g^k |V|)$ time, which is sub-quadratic in $|V|$ when $d_g = o(|V|^{1/k})$, and $O(|V|^2)$ in the worst case.
• The algorithm establishes the existence of $\Omega(d_g^{k-1})$ vertex-disjoint paths in a $\Theta(d_g)^k$ grid, enabling efficient flow routing.
• The framework extends to continuous settings via expansion and grid alignment, preserving the $O(1)$-approximation guarantee for disc robots.