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Reconfiguration of Squares Using a Constant Number of Moves Each

Thijs van der Horst, Maarten Löffler, Tim Ophelders, Tom Peters

TL;DR

This work investigates restricted variants of the problem where square robots are allowed to slide over an arbitrary curve to a new position only a constant number of times each, and shows that the problem remains NP-hard in most cases.

Abstract

Multi-robot motion planning is a hard problem. We investigate restricted variants of the problem where square robots are allowed to slide over an arbitrary curve to a new position only a constant number of times each. We show that the problem remains NP-hard in most cases, except when the squares have unit size and when the problem is unlabeled, i.e., the location of each square in the target configuration is left unspecified.

Reconfiguration of Squares Using a Constant Number of Moves Each

TL;DR

This work investigates restricted variants of the problem where square robots are allowed to slide over an arbitrary curve to a new position only a constant number of times each, and shows that the problem remains NP-hard in most cases.

Abstract

Multi-robot motion planning is a hard problem. We investigate restricted variants of the problem where square robots are allowed to slide over an arbitrary curve to a new position only a constant number of times each. We show that the problem remains NP-hard in most cases, except when the squares have unit size and when the problem is unlabeled, i.e., the location of each square in the target configuration is left unspecified.
Paper Structure (8 sections, 7 theorems, 8 figures, 1 table)

This paper contains 8 sections, 7 theorems, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminairies
  3. Labeled Monotonic Reconfiguration
  4. Unlabeled 1x1
  5. Unlabeled, Big Squares, Monotonic
  6. Bounded, Big Squares, Multiple Moves
  7. Bounded, Labeled 1x1, Multiple Moves
  8. Conclusion

Key Result

Theorem 1

The labeled, monotonic square reconfiguration problem is NP-hard.

Figures (8)

  • Figure 1: The gadgets for Theorem \ref{['thm:manta']}. From left to right: The start gadget, the variable gadget, and the end gadget. Bottom: clause gadget. The green squares represent true, the red ones false. The dotted squares are targets.
  • Figure 2: The gadgets for Theorem \ref{['thm:Hamiltonian']}. In reading order: the merge, split, corner, wire (with a shift), start, and end gadget. For the merge gadget, the two horizontal edges are incoming. For the split gadget, the two horizontal edges are outgoing. Filled squares are starting locations. The dashed squares are target locations. Both the green as well as the yellow squares cover a start as well as a target square.
  • Figure 3: An instance of our Hamiltonian path reduction with a valid movement schedule. Top: The square reconfiguration problem, bottom: The corresponding directed graph.
  • Figure 4: The gadgets for Theorem \ref{['thm:rendez-2move']}. From left to right: Variable gadget, splitter gadget, wire gadget with a corner, and clause gadget. All squares have a target below them, except for the clause checker square.
  • Figure 5: The variable gadget where the yellow latch square can move at most $6$ times. Setting the variable to false (bottom) forces the latch square to move twice. Resetting it takes another two moves. Similarly, setting the variable to true takes three moves, and resetting it takes another three.
  • ...and 3 more figures

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7