Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Efficient Path Planning and Task Allocation Algorithm for Boolean Specifications

Ioana Hustiu, Roozbeh Abolpour, Marius Kloetzer, Cristian Mahulea

TL;DR

This work develops a Petri-net–based framework for simultaneous path planning, task allocation, and Boolean final-state specifications in large multi-robot systems. By proving the constraint matrix is totally unimodular, it replaces NP-hard ILP with tractable LP/MILP relaxations while preserving integrality, enabling scalability to thousands of robots. It extends the TAPF problem to Boolean specifications and introduces a two-stage solver that uses synchronization points to guarantee collision-free execution. Extensive simulations on TAPF, Boolean specifications, and MAPF benchmarks validate strong scalability, competitive runtimes, and formal guarantees, bridging optimization-based TAPF and MAPF approaches for complex, large-scale coordination tasks.

Abstract

This paper addresses path planning and task allocation in multi-robot systems subject to global Boolean specifications defined on the final state. The main contribution is the exploitation of the structural properties of a Petri net model: we prove that the associated constraint matrix is totally unimodular (TU). This property allows relaxing the original Integer Linear Programming (ILP) formulation to a Mixed Integer Linear Programming (MILP) in which all variables are continuous except for those that are corresponding to the atomic propositions in the Boolean specification. This yields a substantial reduction in complexity. In the special case where the specification is a conjunction of atomic propositions of cardinality equal to the team size, i.e., the standard Task-Assignment and Path Finding (TAPF) problem, the formulation reduces to a Linear Programming (LP). Collision-free paths are ensured by introducing intermediate synchronization points only when necessary, while robots move in parallel between them. These structural insights enable a computationally efficient and scalable solution, achieving tractability and safety for large-scale systems with up to 2500 robots.

Efficient Path Planning and Task Allocation Algorithm for Boolean Specifications

TL;DR

This work develops a Petri-net–based framework for simultaneous path planning, task allocation, and Boolean final-state specifications in large multi-robot systems. By proving the constraint matrix is totally unimodular, it replaces NP-hard ILP with tractable LP/MILP relaxations while preserving integrality, enabling scalability to thousands of robots. It extends the TAPF problem to Boolean specifications and introduces a two-stage solver that uses synchronization points to guarantee collision-free execution. Extensive simulations on TAPF, Boolean specifications, and MAPF benchmarks validate strong scalability, competitive runtimes, and formal guarantees, bridging optimization-based TAPF and MAPF approaches for complex, large-scale coordination tasks.

Abstract

This paper addresses path planning and task allocation in multi-robot systems subject to global Boolean specifications defined on the final state. The main contribution is the exploitation of the structural properties of a Petri net model: we prove that the associated constraint matrix is totally unimodular (TU). This property allows relaxing the original Integer Linear Programming (ILP) formulation to a Mixed Integer Linear Programming (MILP) in which all variables are continuous except for those that are corresponding to the atomic propositions in the Boolean specification. This yields a substantial reduction in complexity. In the special case where the specification is a conjunction of atomic propositions of cardinality equal to the team size, i.e., the standard Task-Assignment and Path Finding (TAPF) problem, the formulation reduces to a Linear Programming (LP). Collision-free paths are ensured by introducing intermediate synchronization points only when necessary, while robots move in parallel between them. These structural insights enable a computationally efficient and scalable solution, achieving tractability and safety for large-scale systems with up to 2500 robots.

Paper Structure

This paper contains 12 sections, 4 theorems, 21 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work and Contributions
  3. Problem definition
  4. Unimodular and Totally Unimodular Matrices
  5. Task-Assignment and Path Finding Problem
  6. Extended TAPF Solution Strategy
  7. Simulations
  8. Task-Assignment and Path Finding (TAPF) problem
  9. Boolean-Based Specifications
  10. Comparative Evaluation with MAPF Baselines
  11. Conclusions
  12. Proof of Theorem \ref{['th:TU_final_m']}

Key Result

Theorem 1

Let $\hbox{\boldmath $A$}\in \mathbb{Z}^{m \times n}$ be totally unimodular and let $\hbox{\boldmath $b$}\in \mathbb{Z}^m$ be an integer right-hand side vector. Then every vertex (extreme point) of the polyhedron is integer. Consequently, the LP relaxation of any integer program with constraint matrix $\hbox{\boldmath $A$}$ and integral $\hbox{\boldmath $b$}$ admits an integer optimal solution.

Figures (4)

  • Figure 1: Example of a RMPN.
  • Figure 2: Example of the ht_chantry benchmark environment used for TAPF simulations. Black pixels denote obstacles, while colored markers represent robots and their assigned goal regions for a trial involving 30 robots. Each free pixel corresponds to a place in the underlying RMPN model.
  • Figure 3: Example of warehouse environment used in Boolean-based experiments. Robots start on the left side and must reach central regions that satisfy a randomly generated Boolean formula. Each corridor can host at most one robot at a time.
  • Figure 4: Runtime measured in seconds for room-32-32-4, random-32-32-20, den312d, and ht_chantry maps.

Theorems & Definitions (13)

  • Definition 1
  • Example 1
  • Example 2
  • Remark 1
  • Remark 2
  • Definition 2
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof
  • ...and 3 more