Multi-Agent Path Finding (MAPF)

• Multi-Agent Path Finding (MAPF) is the problem of planning collision-free paths for multiple agents in a shared, graph-based environment.
• Key algorithmic paradigms include search-based techniques like Conflict-Based Search, compilation-based methods using SAT/MIP, and hybrid learning-guided frameworks.
• MAPF is vital in real-world applications such as automated warehouse systems, robotic fleets, and air traffic control, addressing dynamic constraints and high-density scenarios.

Multi-Agent Path Finding (MAPF) is the problem of computing collision-free paths for multiple agents navigating simultaneously in a shared environment, typically represented as a graph. Each agent must move from a specified start location to a designated goal while ensuring spatiotemporal separation from other agents. MAPF has become central to a wide range of domains, including automated warehouse management, robot fleets, air traffic control, and digital games. Variants address lifelong settings (continuous reassignment of goals), real-world dynamics, agent rationality, and rich environmental constraints. Recent research demonstrates a diverse algorithmic landscape, including optimal and suboptimal discrete solvers, continuous and any-angle extensions, hybrid search+compilation methods, learning-based and distributed approaches, and formulations accounting for execution uncertainty, high density, or strategic behavior.

1. Standard Problem Formulation and Complexity

The canonical MAPF instance is defined by a tuple $$(k, G=(V,E), s: [1..k]\rightarrow V, t: [1..k]\rightarrow V)$$, where $k$ is the number of agents, $G$ is an undirected graph (discrete workspace), and $s_i$, $t_i$ are the start and goal vertices for agent $i$ (Shabalin et al., 2024). Time is discretized; at every step, an agent may move to an adjacent vertex or wait in place. A solution comprises $k$ time-indexed paths, one per agent, such that:

• Vertex collision: $\forall t$, no two agents occupy the same vertex.
• Edge collision: $\forall t$, no two agents swap endpoints along an edge at the same time.

The two most common optimization objectives are sum-of-costs (SOC), $\mathrm{SOC} = \sum_{i=1}^k |\pi_i|$, and makespan, $\mathrm{MS} = \max_i |\pi_i|$.

MAPF is NP-hard under both measures, even for small networks and grid-like environments (Surynek, 2021). This computational barrier is exacerbated in high-density or lifelong/online variants.

2. Algorithmic Paradigms

MAPF solvers have evolved along several axes, leveraging discrete search, constraint-programming encodings, geometric extensions, and learning-based techniques.

Search-based approaches: Conflict-Based Search (CBS) and its variants dominate optimal discrete MAPF, employing a two-level scheme: high-level conflict resolution by constraint branching, and low-level single-agent planning (Andreychuk et al., 2019, Vainshtein et al., 2022). Subdimensional expansion (M*, BPM*) exploits the existence of uncoupled agents, expanding only higher-dimensional subspaces as needed (2207.14657). Continuous time and any-angle settings adapt CBS with safe-interval–path-planning subroutines (CCBS, TO-AA-SIPP) (Andreychuk et al., 2019, Yakovlev et al., 2024).

Compilation-based solvers: The problem is reduced to ASP, SAT, or MIP instances, benefiting from decades of generalized solver development (Surynek, 2021). SAT encodings in particular (e.g., MDD-SAT) excel for large agent counts on grid maps, while MIP-based branch-and-cut optimally handles sum-of-costs for dense or highly constrained instances.

Hybrid and advanced frameworks: Learning-guided compilation and search+compilation hybrids (e.g.,