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Constrained Multi-Agent Path Finding on Directed Graphs

Stefano Ardizzoni, Luca Consolini, Marco Locatelli, Irene Saccani

TL;DR

The paper introduces constrained variants of MP and MAPF on directed graphs, where occupancy constraints are captured by ASC constraint sets $\\\mathcal{C}$ defined via pairs $(S,k)$. It proves NP-hardness of feasibility for both $\\mathcal{C}$-MP and $\\mathcal{C}$-MAPF and develops a reduction framework that maps $\\mathcal{C}$-MAPF to standard MAPF by selecting an independent set $W$ and constructing a reduced graph $G_W$; solutions on $G_W$ can be lifted to solutions on the original graph, though the reduction is not exact. The authors show that maximizing $|W|$ (the $\\mathcal{C}$-MIS problem) is strongly NP-hard and provide polynomial-time heuristics to compute suboptimal $W$. They validate the approach with real-world warehouse layouts and square-grid experiments, illustrating practical applicability and highlighting the trade-off between reduced problem complexity and potential loss of feasible solutions. The work points to future directions for more robust reductions and more efficient MIS heuristics to expand the class of instances amenable to MAPF-based planning under occupancy constraints.

Abstract

We discuss C-MP and C-MAPF, generalizations of the classical Motion Planning (MP) and Multi-Agent Path Finding (MAPF) problems on a directed graph G. Namely, we enforce an upper bound on the number of agents that occupy each member of a family of vertex subsets. For instance, this constraint allows maintaining a safety distance between agents. We prove that finding a feasible solution of C-MP and C-MAPF is NP-hard, and we propose a reduction method to convert them to standard MP and MAPF. This reduction method consists in finding a subset of nodes W and a reduced graph G/W, such that a solution of MAPF on G/W provides a solution of C-MAPF on G. Moreover, we study the problem of finding W of maximum cardinality, which is strongly NP-hard.

Constrained Multi-Agent Path Finding on Directed Graphs

TL;DR

The paper introduces constrained variants of MP and MAPF on directed graphs, where occupancy constraints are captured by ASC constraint sets defined via pairs . It proves NP-hardness of feasibility for both -MP and -MAPF and develops a reduction framework that maps -MAPF to standard MAPF by selecting an independent set and constructing a reduced graph ; solutions on can be lifted to solutions on the original graph, though the reduction is not exact. The authors show that maximizing (the -MIS problem) is strongly NP-hard and provide polynomial-time heuristics to compute suboptimal . They validate the approach with real-world warehouse layouts and square-grid experiments, illustrating practical applicability and highlighting the trade-off between reduced problem complexity and potential loss of feasible solutions. The work points to future directions for more robust reductions and more efficient MIS heuristics to expand the class of instances amenable to MAPF-based planning under occupancy constraints.

Abstract

We discuss C-MP and C-MAPF, generalizations of the classical Motion Planning (MP) and Multi-Agent Path Finding (MAPF) problems on a directed graph G. Namely, we enforce an upper bound on the number of agents that occupy each member of a family of vertex subsets. For instance, this constraint allows maintaining a safety distance between agents. We prove that finding a feasible solution of C-MP and C-MAPF is NP-hard, and we propose a reduction method to convert them to standard MP and MAPF. This reduction method consists in finding a subset of nodes W and a reduced graph G/W, such that a solution of MAPF on G/W provides a solution of C-MAPF on G. Moreover, we study the problem of finding W of maximum cardinality, which is strongly NP-hard.
Paper Structure (11 sections, 14 theorems, 22 equations, 14 figures, 1 algorithm)

This paper contains 11 sections, 14 theorems, 22 equations, 14 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Definition
  3. MAPF and MP with additional constraints
  4. Complexity of the motion planning problem with additional constraints
  5. Reduction of $\mathcal{C}$-MAPF into MAPF
  6. Finding an independent set of nodes with largest cardinality
  7. Heuristics for the $\mathcal{C}$-MIS problem
  8. Experimental results
  9. Real-life applications examples of $\mathcal{C}$-MAPF
  10. $\mathcal{C}$-MIS problem on square grid graphs
  11. Conclusion and future works

Key Result

Proposition 2.1

A constraint set $\mathcal{C}$ is an ASC if and only if there exists a family ${\cal Q}$ of pairs $(S,k)$ such that $\mathcal{C}=\mathcal{C}_{\cal Q}$.

Figures (14)

  • Figure 1: Examples of $C$-MAPF instances.
  • Figure 2: Reduced graphs associated to examples of Figure 1.
  • Figure 3: Reduction of the 3-SAT instance $(x_1\vee \bar{x}_2) \wedge x_3 \wedge (\bar{x}_1\vee x_2 \vee \bar{x}_3)$ to an $\mathcal{C}$-MP instance.
  • Figure 4: Component $\{o_i, x_i,\bar{x}_i\}$.
  • Figure 5: Component associated to the $k$ clauses.
  • ...and 9 more figures

Theorems & Definitions (41)

  • Definition 1.1
  • Definition 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.1
  • proof
  • Remark 2.1
  • Definition 2.4
  • Definition 2.5
  • ...and 31 more