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Tractability Frontiers in Multi-Robot Coordination and Geometric Reconfiguration

Tzvika Geft, Dan Halperin, Yonatan Nakar

TL;DR

The paper studies Monotone Sliding Reconfiguration ($MSR$) for labeled, interior-disjoint objects in the plane, showing $MSR$ is NP-hard even without obstacles and under line-separable conditions, while identifying a structural tractable frontier via _spannable by a forest of full binary trees_ ($SFFBT$) that yields efficient algorithms for unit discs and 2D grids. It establishes a core reduction from Pivot Scheduling to $MSR$, and builds a bridge to pebble motion graphs (PMG) to transfer tractability under the SFFBT condition. The authors also develop efficient tests for spannability in geometric settings using endpoint-free spaces, plane arrangements, and Courcelle’s theorem, and present a restricted yet always-solvable thin-polyomino case. Overall, the work demarcates sharp hardness boundaries and practical solvable instances, pushing toward denser, more realistic multi-robot configurations."

Abstract

We study the Monotone Sliding Reconfiguration (MSR) problem, in which $\textit{labeled}$ pairwise interior-disjoint objects in a planar workspace need to be brought $\textit{one by one}$ from their initial positions to given target positions, without causing collisions. That is, at each step only one object moves to its respective target, where it stays thereafter. MSR is a natural special variant of Multi-Robot Motion Planning (MRMP) and related reconfiguration problems, many of which are known to be computationally hard. A key question is identifying the minimal mitigating assumptions that enable efficient algorithms for such problems. We first show that despite the monotonicity requirement, MSR remains a computationally hard MRMP problem. We then provide additional hardness results for MSR that rule out several natural assumptions. For example, we show that MSR remains hard without obstacles in the workspace. On the positive side, we introduce a family of MSR instances that always have a solution through a novel structural assumption pertaining to the graphs underlying the start and target configuration -- we require that these graphs are spannable by a forest of full binary trees (SFFBT). We use our assumption to obtain efficient MSR algorithms for unit discs and 2D grid settings. Notably, our assumption does not require separation between start/target positions, which is a standard requirement in efficient and complete MRMP algorithms. Instead, we (implicitly) require separation between $\textit{groups}$ of these positions, thereby pushing the boundary of efficiently solvable instances toward denser scenarios.

Tractability Frontiers in Multi-Robot Coordination and Geometric Reconfiguration

TL;DR

The paper studies Monotone Sliding Reconfiguration () for labeled, interior-disjoint objects in the plane, showing is NP-hard even without obstacles and under line-separable conditions, while identifying a structural tractable frontier via _spannable by a forest of full binary trees_ () that yields efficient algorithms for unit discs and 2D grids. It establishes a core reduction from Pivot Scheduling to , and builds a bridge to pebble motion graphs (PMG) to transfer tractability under the SFFBT condition. The authors also develop efficient tests for spannability in geometric settings using endpoint-free spaces, plane arrangements, and Courcelle’s theorem, and present a restricted yet always-solvable thin-polyomino case. Overall, the work demarcates sharp hardness boundaries and practical solvable instances, pushing toward denser, more realistic multi-robot configurations."

Abstract

We study the Monotone Sliding Reconfiguration (MSR) problem, in which pairwise interior-disjoint objects in a planar workspace need to be brought from their initial positions to given target positions, without causing collisions. That is, at each step only one object moves to its respective target, where it stays thereafter. MSR is a natural special variant of Multi-Robot Motion Planning (MRMP) and related reconfiguration problems, many of which are known to be computationally hard. A key question is identifying the minimal mitigating assumptions that enable efficient algorithms for such problems. We first show that despite the monotonicity requirement, MSR remains a computationally hard MRMP problem. We then provide additional hardness results for MSR that rule out several natural assumptions. For example, we show that MSR remains hard without obstacles in the workspace. On the positive side, we introduce a family of MSR instances that always have a solution through a novel structural assumption pertaining to the graphs underlying the start and target configuration -- we require that these graphs are spannable by a forest of full binary trees (SFFBT). We use our assumption to obtain efficient MSR algorithms for unit discs and 2D grid settings. Notably, our assumption does not require separation between start/target positions, which is a standard requirement in efficient and complete MRMP algorithms. Instead, we (implicitly) require separation between of these positions, thereby pushing the boundary of efficiently solvable instances toward denser scenarios.

Paper Structure

This paper contains 16 sections, 18 theorems, 1 equation, 10 figures.

Table of Contents

  1. Introduction
  2. Hardness results
  3. Establishing the hardness of MSR
  4. Relaxing assumptions of tractable cases
  5. Hardness without obstacles
  6. A monotone solution in pebble motion graphs
  7. Solving an instance with a spannable motion graph
  8. Hardness of deciding whether a motion graph is spannable
  9. Application in geometric settings
  10. Preliminaries and background
  11. Constructing the motion graph
  12. Efficiently deciding whether the motion graph is spannable
  13. A restricted case that always has a monotone solution
  14. Conclusion
  15. Missing proofs
  16. ...and 1 more sections

Key Result

theorem thmcountertheorem

MSR is NP-hard in each of the following variants: and the following obstacle-free line-separable variants:

Figures (10)

  • Figure 1: MSR instances. Filled (resp. unfilled) squares are start (resp. target) positions. (a) A "No" instance, as one of the squares must make two moves. (b) An instance that can be solved by moving the squares in the order $b,a,c$ and no other order. (c) A modification of (b) to a well-formed instance (see text).
  • Figure 2: Two start configurations featuring disks with little to no separation between them, which are allowed under our assumption. The arrows (right) illustrate the core concept of SFFBT (see text).
  • Figure 3: The MSR instance $M$ for the Pivot Scheduling instance with the before constraints $X_1 = \{x_1, x'_1\}, Y_1=\{y_1\}, X_2 = \{x_2\}, Y_2 = \{y_2, y'_2\}, X_3 = \{x_3, x'_3\}, Y_3 = \{y_3\}$ and after constraints $\mathcal{C}=\{\{y_1,y_2,x_3\}, \{x_1,y'_2,x'_3\}, \{x'_1,x_2,y_3\}\}$. Obstacles appear in gray. The start and target positions are the filled and unfilled colored squares, respectively. The start and target positions of the $R(X_i)$ (resp. $R(Y_i)$) robots are green (resp. red). Labels distinguish between robots having the same color. The path $P_{B\xspace,A\xspace}$ (blue) is shown for the solution $B\xspace{} = \{y_1, x_2, y_3\}, A\xspace{} = \{x_1,x'_1, y_2,y'_2, x_3,x'_3\}$.
  • Figure 4: The instance $M'_1$, obtained from the instance $M$ (\ref{['fig:main-construction']}), for $\varepsilon = 1/3$.
  • Figure 5: The instance $M'_2$, obtained from the instance $M$ (\ref{['fig:main-construction']}). For clarity, we label the robots associated with $X_i$, which are highlighted using yellow-shaded cells, and omit most other labels.
  • ...and 5 more figures

Theorems & Definitions (34)

  • theorem thmcountertheorem
  • lemma thmcounterlemma
  • theorem thmcountertheorem
  • proof
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • ...and 24 more