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Well-Connected Set and Its Application to Multi-Robot Path Planning

Teng Guo, Jingjin Yu

TL;DR

It is established that computing an LWCS is NP-complete, and optimal and near-optimal LWCS algorithms are developed, with the near-optimal algorithm targeting large maps.

Abstract

Parking lots and autonomous warehouses for accommodating many vehicles/robots adopt designs in which the underlying graphs are \emph{well-connected} to simplify planning and reduce congestion. In this study, we formulate and delve into the \emph{largest well-connected set} (LWCS) problem and explore its applications in layout design for multi-robot path planning. Roughly speaking, a well-connected set over a connected graph is a set of vertices such that there is a path on the graph connecting any pair of vertices in the set without passing through any additional vertices of the set. Identifying an LWCS has many potential high-utility applications, e.g., for determining parking garage layout and capacity, as prioritized planning can be shown to be complete when start/goal configurations belong to an LWCS. In this work, we establish that computing an LWCS is NP-complete. We further develop optimal and near-optimal LWCS algorithms, with the near-optimal algorithm targeting large maps. A complete prioritized planning method is given for planning paths for multiple robots residing on an LWCS.

Well-Connected Set and Its Application to Multi-Robot Path Planning

TL;DR

It is established that computing an LWCS is NP-complete, and optimal and near-optimal LWCS algorithms are developed, with the near-optimal algorithm targeting large maps.

Abstract

Parking lots and autonomous warehouses for accommodating many vehicles/robots adopt designs in which the underlying graphs are \emph{well-connected} to simplify planning and reduce congestion. In this study, we formulate and delve into the \emph{largest well-connected set} (LWCS) problem and explore its applications in layout design for multi-robot path planning. Roughly speaking, a well-connected set over a connected graph is a set of vertices such that there is a path on the graph connecting any pair of vertices in the set without passing through any additional vertices of the set. Identifying an LWCS has many potential high-utility applications, e.g., for determining parking garage layout and capacity, as prioritized planning can be shown to be complete when start/goal configurations belong to an LWCS. In this work, we establish that computing an LWCS is NP-complete. We further develop optimal and near-optimal LWCS algorithms, with the near-optimal algorithm targeting large maps. A complete prioritized planning method is given for planning paths for multiple robots residing on an LWCS.
Paper Structure (13 sections, 8 theorems, 6 figures, 2 tables, 3 algorithms)

This paper contains 13 sections, 8 theorems, 6 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Well-Connected Set
  4. Theoretic Study
  5. Algorithms for finding well-connected sets
  6. Algorithm for Finding a MWCS
  7. Exact Search-Based Algorithm
  8. Applications in multi-robot navigation
  9. Experiments
  10. Grid Experiments
  11. Benchmark Maps
  12. Evaluations of MRPP
  13. Conclusions

Key Result

Proposition III.1

If $M$ is a WCS, for any $M'\subseteq M,v\in M'$, $v$ is not an articulation point of the subgraph induced by $V-M'+\{v\}$.

Figures (6)

  • Figure 1: Examples of well-formed infrastructures. (a) Amazon fulfillment warehouse. (b) A typical parking lot.
  • Figure 2: (a) The green cells form a WCS. Any robot parked at one of these cells does not block others' move. (b) An example of a non-WCS. Retrieving one robot, $A$, requires moving some other robots. (c) An example of a SWCS. $B$ has no access to a robot at $C$ without moving others.
  • Figure 3: The graph derived from the 3SAT instance $(x_1\vee x_2\vee x_3)\wedge(x_2\vee \Bar{x}_3\vee \Bar{x}_4) \wedge (\Bar{x}_1\vee x_3\vee x_4)\wedge(x_1\vee\Bar{x}_2\vee\Bar{x}_4)$. The green vertices form a LWCS of size 12. By setting the literals of green vertices to true, the 3SAT instance is satisfied.
  • Figure 4: Examples of the computed MWCS (colored in green) in different 4-connected grid maps. (a) den312d. (b) ht$\_$chantry. (c) Shanghai$\_$0$\_$256. (d) lak503d.Zoom in on the digital version to see more details.
  • Figure 5: Experimental results for map orz201d, including computation time, success rate, makespan optimality, and soc optimality, for HCA, PIBT, and the proposed method.
  • ...and 1 more figures

Theorems & Definitions (20)

  • Definition : Well-Connected Set (WCS)
  • Definition : Maximal Well-Connected Set (MWCS)
  • Definition : Largest Well-Connected Set (LWCS)
  • Definition : Well-Connected Path (WCP)
  • Definition : Path Efficiency Ratio
  • Proposition III.1
  • Proposition III.2
  • proof
  • Theorem III.1: Intractability
  • proof
  • ...and 10 more