Graph Coloring to Reduce Computation Time in Prioritized Planning

TL;DR

This work tackles reducing computation time in prioritized planning (PP) for multi-agent MAPF by modeling agent interactions with a directed acyclic graph (DAG) and showing that the longest path length $L$ governs PP time. It maps the minimization of $L$ to a graph-coloring problem on the coupling graph and provides a polynomial-time decentralized greedy coloring algorithm, where the number of colors equals the number of computation levels in the coupling DAG. The approach is validated in a multi-agent trajectory planning setting for connected autonomous vehicles (CAVs) at intersections, achieving a reduction in mamp computation time by over $57.9\%$ at the cost of a modest $26.2\%$ increase in trajectory cost, highlighting benefits in sparse coupling graphs and scalability. This decentralized coloring strategy offers a robust, low-communication method to accelerate PP-based planning in large networks, with potential applicability across domains requiring prioritized, parallelizable computations.

Abstract

Distributing computations among agents in large networks reduces computational effort in multi-agent path finding (MAPF). One distribution strategy is prioritized planning (PP). In PP, we couple and prioritize interacting agents to achieve a desired behavior across all agents in the network. We characterize the interaction with a directed acyclic graph (DAG). The computation time for solving MAPF problem using PP is mainly determined through the longest path in this DAG. The longest path depends on the fixed undirected coupling graph and the variable prioritization. The approaches from literature to prioritize agents are numerous and pursue various goals. This article presents an approach for prioritization in PP to reduce the longest path length in the coupling DAG and thus the computation time for MAPF using PP. We prove that this problem can be mapped to a graph-coloring problem, in which the number of colors required corresponds to the longest path length in the coupling DAG. We propose a decentralized graph-coloring algorithm to determine priorities for the agents. We evaluate the approach by applying it to multi-agent motion planning (MAMP) for connected and automated vehicles (CAVs) on roads using, a variant of MAPF.

Figures (9)

• Figure 1: Our framework of pp which runs every time step. $\glslink{sym:stateAgent}{^{(i)}_{k}}$ is the state, $$ is the undirected coupling graph, $$ is the directed coupling graph, $\glslink{sym:prediction}{ \IfNoValueTF{-NoValue-} { \bm{x}_{\cdot \,\vert\, k} } {\IfNoValueTF{-NoValue-} { \bm{x}_{\cdot \,\vert\, k}^{(-NoValue-)} } { \bm{x}_{\cdot \,\vert\, k}^{(-NoValue- \,\vert\, -NoValue-)} } } }^{(i)}$ is the prediction of agent $i$, $\glslink{sym:prediction}{ \IfNoValueTF{-NoValue-} { \bm{x}_{\cdot \,\vert\, k} } {\IfNoValueTF{-NoValue-} { \bm{x}_{\cdot \,\vert\, k}^{(-NoValue-)} } { \bm{x}_{\cdot \,\vert\, k}^{(-NoValue- \,\vert\, -NoValue-)} } } }^{(j \,\vert\, i)}$ is the the prediction of agent $j$ in agent $i$.
• Figure 2: Example of problem solution process
• Figure 3: Coupling graph for eight-lane intersection with eight vehicles. For each direction of the intersection, one vehicle turns right and one vehicle moves straight, which is indicated by dots. The undirected coupling graph is based on the potential collisions between vehicles.
• Figure 4: Computation time using $_\text{color}$ (our approach) compared to other prioritizations in the experiment starting from the initial states shown in \ref{['fig:eval_scenario_coupling']}.
• Figure 5: Number of prioritizations for the coupling graph in \ref{['fig:eval_scenario_coupling']} that result in a coupling dag with the given number of computation levels $$.

Theorems & Definitions (15)

• Definition 1: Prediction
• Definition 2: Prediction Consistency
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Lemma 1
• proof
• Theorem 1
• proof