Layered LA-MAPF: a decomposition of large agent MAPF instance to accelerate solving without compromising solvability

TL;DR

Layered LA-MAPF is introduced, a method that decomposes a MAPF instance involving geometric agents into clusters, and then further decomposes each cluster into levels, and aims to reduce time complexity when solving LA-MAPF problems.

Abstract

Multi-Agent Path Finding (MAPF) has been widely studied in recent years. However, most existing MAPF algorithms assume that an agent occupies only a single grid in a grid-based map. This assumption limits their applicability in many real-world domains where agents have geometric shapes, rather than being point-like. Such agents, which can occupy multiple cells simultaneously, are referred to as ``large'' agents. When considering the shape and size of agents in MAPF, the computational complexity increases significantly as the number of agents grows, primarily due to the increased overhead in conflict detection between geometric agents. In this paper, we propose two types of subproblems for the LA-MAPF (Large-Agent MAPF) problem: \textbf{cluster} (which has no constraints on the order of solution) and \textbf{level} (which imposes constraints on the solution order). We introduce \textbf{Layered LA-MAPF}, a method that decomposes a MAPF instance involving geometric agents into clusters, and then further decomposes each cluster into levels. This approach aims to reduce time complexity when solving LA-MAPF problems. Our results demonstrate the performance of our method as the number of agents increases across various maps, and how it accelerates LA-MAPF methods, such as LA-CBS and LA-LaCAM. Experiments show that our LA-MAPF method with instance decomposition \textbf{halves the time cost (reducing from an average of 40s to 20s) and triples the success rate (from an average of 0.27 to 0.80)} in finding a solution within 60 seconds. To facilitate further research, we have made the source code for Layered LA-MAPF publicly available at \url{https://github.com/JoeYao-bit/LayeredMAPF/algorithm/LA-MAPF}.

Figures (21)

• Figure 1: Figure A and B show an example about two kinds of subproblem (cluster and level) in MAPF that can be solved independently. $S_{i}$ and $G_{i}$ means start locations and target locations about agent $a_i$ respectively. Grey grids represent obstacles and white grids represent passable grids. For MAPF instance in Figure A, we can solve $a_1$ and $a_2$ separately and have no limitation in order of solve. If we solve $a_1$ first, the final solution might be $a_1$ = $\{(1,2)\rightarrow(1,1)\rightarrow(1,0)\}$, and $a_2$ = $\{(0,1)\rightarrow(0,1)\rightarrow(1,1)\rightarrow(2,1)\}$. If we solve $a_2$ first, the final solution might be $a_1$ = $\{(1,2)\rightarrow(1,2)\rightarrow(1,1)\rightarrow(1,0)\}$, and $a_2$ = $\{(0,1)\rightarrow(1,1)\rightarrow(2,1)\}$. So we say MAPF instance in Figure A can be decompose into two clusters, cluster 1 = $\{a_1\}$, cluster 2 = $\{a_2\}$. For MAPF instance in Figure B, we can solve $a_1$ and $a_2$ separately but there is a limitation in order of solve. If we solve $a_1$ first, the solution of $a_1$ = $\{(1,2)\rightarrow(1,1)\}$, then $a_2$ find no conflict free solution while considering $a_1$ as dynamic obstacles. If we solve $a_2$ first, the final solution might be $a_1$ = $\{(1,2)\rightarrow(1,2)\rightarrow(1,1)\}$, and $a_2$ = $\{(0,1)\rightarrow(1,1)\rightarrow(2,1)\}$. So we say MAPF instance in Figure B can only be decompose into two levels, level 1 = $\{a_2\}$, level 2 = $\{a_1\}$. Agents in level 1 must be solved before level 2 to ensure the solvability of MAPF instance decomposition.
• Figure 2: These figures illustrate the complete process of decomposing an LA-MAPF instance, which includes decomposing agents into initial clusters , further decomposing the initial clusters into smaller clusters, and then decomposing the clusters into initial levels, followed by decomposing the initial levels into smaller levels. For more details, please refer to Section Methodology.
• Figure 3: Figures A and B illustrate how we check whether an agent collides with obstacle grids or another agent. Figure A depicts a circular agent, while Figure B shows a circular agent and a rectangular agent, with the grids they occupy filled with slashes. It is evident that in Figure A, the rectangle agent collides with the obstacle grids, and in Figure B, the circular and rectangular agents collide with each other.
• Figure 4: Figure A shows a circular agent and a block agent, which occupy 9 grids and 2 grids respectively. Figure B presents the subgraph of the circular agent, while Figure C shows the subgraph of the block agent. Each arrow indicates a robot state, represented as a tuple of location and orientation.
• Figure 5: Figure A shows a circular agent and a block agent having vertex conflicts. The conflict arises because their occupied grids overlap at the same timestep. Figure B shows two block agents having a transfer conflict. This conflict occurs when both agents are transitioning from one orientation to another simultaneously, and their occupied grids overlap during the transfer.

Theorems & Definitions (27)

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