Conflict-Based Search for Multi-Agent Path Finding with Elevators

Abstract

This paper investigates a problem called Multi-Agent Path Finding with Elevators (MAPF-E), which seeks conflict-free paths for multiple agents from their start to goal locations that may locate on different floors, and the agents can use elevators to travel between floors. The existence of elevators complicates the interaction among the agents and introduces new challenges to the planning. On the one hand, elevators can cause many conflicts among the agents due to its relatively long traversal time across floors, especially when many agents need to reach a different floor. On the other hand, the planner has to reason in a larger state space including the states of the elevators, besides the locations of the agents.

Figures (9)

• Figure 1: (a) The motivation of MAPF-E. Robots move in a 3D warehouse, while inter-layer transitions are enabled by the elevators (the orange structures). The picture is from ZS Robotics zsrobotics2025. (b) A toy example of MAPF-E, which seeks start-goal paths for the agents. All elevator edge costs are 2, and the black squares represent obstacles. (c) At time $t=2$, agent $i$ is using the elevator, and agent $j$ needs to wait at a vertex outside the elevator.
• Figure 2: Agent $i$ (blue) and $j$ (green) need to use the same elevator $k$ to reach their goals. An elevator conflict arises when agent $j$ attempts to enter the elevator while it is in the occupied or resetting state, i.e., when $t_s^j \in [t_s^i,\, t_g^i + \triangle]$.
• Figure 3: Each node $(v,t)$ in the MDD means the agent can optimally reach the goal via vertex $v$ at time $t$. For instance, in $MDD^i_2$, at $t=1$, the agent can reach either $B2$ or $C1$, resulting in two nodes at this layer. In the joint MDD, each node must correspond to a conflict-free partial path. For instance, in $MDD^{ij}_3$ at $t=1$, a conflict would occur if agents $i$ and $j$ both visited $B2$ simultaneously. So the joint MDD at $t=1$ contains only the node $((C3,1),(B2,1))$.
• Figure 4: (a) A toy example of MAPF-E, which seeks start-goal paths for the agents. The colors indicates the agents. The traversal time of any elevator edge is $t_{floor} = 1$. (b) Inefficiency of resolving the elevator conflict by single-timestep constraint in CBS, which requires expanding the CT at least $\min\{(t^j_s+t^j_o+t^j_r-t^i_s+1),\ (t^i_s+t^i_o+t^i_r-t^j_s+1)\}$ times. (c) By using our EC, the elevator conflict can be resolved with a single expansion.
• Figure 5: (a) A toy example of MAPF-E, where agents (green and blue) move from start to goal, black blocks denote obstacles, and each elevator edge has a traversal time of $t_{floor}=1$. (b) shows that when constructing joint MDD by the classical method, an elevator conflict occurring at $t=3$ cannot be detected, because the nodes in the traditional MDD do not contain elevator-related information and can not capture the state of elevator; (c) shows that based on the MDD-E, the elevator conflict can be successfully detected during the construction of the joint MDD-E.

Theorems & Definitions (5)

• Definition 1: Elevator Conflict
• Theorem 1
• proof
• Remark 1
• Example 1