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Introducing Delays in Multi-Agent Path Finding

Justin Kottinger, Tzvika Geft, Shaull Almagor, Oren Salzman, Morteza Lahijanian

TL;DR

This work addresses plan repair for multi-agent path finding when delays occur by introducing a minimal set of delays to the existing plan (ACID). It proves ACID is $ mathsf{NP}$-complete and $ mathsf{APX}$-hard, and then leverages a MAPF-based reduction to Constrained Graphs (CG) and Improved Constrained Graphs (ICG) to enable efficient solving with standard MAPF solvers such as CBS. Empirical results show that solving ACID on CG/ICG scales to up to 1000 agents and yields faster repairs with smaller added delays compared to replanning on the original graph, with heuristic solvers performing well in more delay-heavy scenarios. The findings demonstrate practical, explainable, and scalable plan repair capabilities for large-scale MAPF deployments under delays, and provide a foundation for robust, centralized delay-aware coordination.

Abstract

We consider a Multi-Agent Path Finding (MAPF) setting where agents have been assigned a plan, but during its execution some agents are delayed. Instead of replanning from scratch when such a delay occurs, we propose delay introduction, whereby we delay some additional agents so that the remainder of the plan can be executed safely. We show that finding the minimum number of additional delays is APX-Hard, i.e., it is NP-Hard to find a $(1+\varepsilon)$-approximation for some $\varepsilon>0$. However, in practice we can find optimal delay-introductions using Conflict-Based Search for very large numbers of agents, and both planning time and the resulting length of the plan are comparable, and sometimes outperform the state-of-the-art heuristics for replanning.

Introducing Delays in Multi-Agent Path Finding

TL;DR

This work addresses plan repair for multi-agent path finding when delays occur by introducing a minimal set of delays to the existing plan (ACID). It proves ACID is -complete and -hard, and then leverages a MAPF-based reduction to Constrained Graphs (CG) and Improved Constrained Graphs (ICG) to enable efficient solving with standard MAPF solvers such as CBS. Empirical results show that solving ACID on CG/ICG scales to up to 1000 agents and yields faster repairs with smaller added delays compared to replanning on the original graph, with heuristic solvers performing well in more delay-heavy scenarios. The findings demonstrate practical, explainable, and scalable plan repair capabilities for large-scale MAPF deployments under delays, and provide a foundation for robust, centralized delay-aware coordination.

Abstract

We consider a Multi-Agent Path Finding (MAPF) setting where agents have been assigned a plan, but during its execution some agents are delayed. Instead of replanning from scratch when such a delay occurs, we propose delay introduction, whereby we delay some additional agents so that the remainder of the plan can be executed safely. We show that finding the minimum number of additional delays is APX-Hard, i.e., it is NP-Hard to find a -approximation for some . However, in practice we can find optimal delay-introductions using Conflict-Based Search for very large numbers of agents, and both planning time and the resulting length of the plan are comparable, and sometimes outperform the state-of-the-art heuristics for replanning.
Paper Structure (16 sections, 3 theorems, 6 figures, 4 tables)

This paper contains 16 sections, 3 theorems, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Computational Complexity of ACID
  4. MAPF Formulation of ACID
  5. Constrained Graph
  6. Improved Constrained Graph
  7. Experimental Evaluation
  8. Experimental Setup -- Single Delay
  9. Computation Time
  10. Success Rate
  11. Added SOC
  12. Experimental Setup – Multiple Delays
  13. Conclusion
  14. Technical Appendix
  15. Proof of Theorem \ref{['thm:ACID_NP_complete']}
  16. ...and 1 more sections

Key Result

Lemma 1

Consider an ACID instance with plan $P=\{\pi_1,\ldots,\pi_n\}$ and budget $D$. If the instance is solvable, then it is also solvable with budget $D'=(n-1) \cdot \|P\|$.

Figures (6)

  • Figure 1: Setting of (a) \ref{['xmp:postpone']} and (b) \ref{['xmp:multiple_delays']}. Straight lines depict the original plan, and curved edges represent the plan when Red agent is delayed at time 0. (a) It is better to have Green agent delay at time 3. (b) it is better to have Green agent delay for two timesteps.
  • Figure 2: An input graph for the reduction with $C=3$. Observe that the graph can be colored with sum $3$, by $\chi(x)=\chi(w)=0$, $\chi(y)=1$ and $\chi(z)=2$. Note that for clarity, we use $x,y,z$ and $w$ and not $1, \ldots, 4$ (as is done in the reduction) to name the vertices.
  • Figure 3: Reduction output. Each agent is represented by a path (e.g., $x$ is the blue path, also distinguished by arrow types). The complete output has $C+1=4$ blocks.
  • Figure 4: (a) A Constrained Graph and (b) an Improved Constrained Graph for an MAPF plan with three agents.
  • Figure 5: MAPF maps used in the experimental evaluations.
  • ...and 1 more figures

Theorems & Definitions (13)

  • Example 1: Postponing delays
  • Example 2: Long delays
  • Lemma 1
  • proof
  • Remark 1: Encoding of the budget $D$
  • Theorem 1
  • proof : Proof sketch
  • Remark 2: ACID variants
  • Theorem 2
  • Remark 3: Upper bound on delays
  • ...and 3 more