A Minor-Testing Approach for Coordinated Motion Planning with Sliding Robots

TL;DR

The paper studies Coordinated Sliding-Motion Planning (CSMP) on undirected graphs where $k$ robots slide along free paths in serial moves, seeking a schedule of at most $\ell$ moves to reach designated destinations. It introduces a novel topological-minor representation of optimal solutions, enabling fixed-parameter tractability: CSMP is FPT by the number of robots $k$, and Planar-CSMP-1 is FPT by the makespan $\ell$ (despite NP-hardness in general). Key techniques include shortening degree-2 paths to bound makespan, havens to untangle interactions, and encoding solutions as rooted topological minors; planarity is exploited via irrelevant-edge contractions and MSO/Courcelle-based model checking. The results unify graph-theoretic and logical methods to address motion-planning reconfiguration, with implications for scalable planning in planar and sparse settings and for topological-minor-based encodings of complex reconfiguration problems.

Abstract

We study a variant of the Coordinated Motion Planning problem on undirected graphs, referred to herein as the \textsc{Coordinated Sliding-Motion Planning} (CSMP) problem. In this variant, we are given an undirected graph $G$, $k$ robots $R_1,\dots,R_k$ positioned on distinct vertices of $G$, $p\leq k$ distinct destination vertices for robots $R_1,\dots,R_p$, and $\ell \in \mathbb{N}$. The problem is to decide if there is a serial schedule of at most $\ell$ moves (i.e., of makespan $\ell$) such that at the end of the schedule each robot with a destination reaches it, where a robot's move is a free path (unoccupied by any robots) from its current position to an unoccupied vertex. The problem is known to be NP-hard even on full grids. It has been studied in several contexts, including coin movement and reconfiguration problems, with respect to feasibility, complexity, and approximation. Geometric variants of the problem, in which congruent geometric-shape robots (e.g., unit disk/squares) slide or translate in the Euclidean plane, have also been studied extensively. We investigate the parameterized complexity of CSMP with respect to two parameters: the number $k$ of robots and the makespan $\ell$. As our first result, we present a fixed-parameter algorithm for CSMP parameterized by $k$. For our second result, we present a fixed-parameter algorithm parameterized by $\ell$ for the special case of CSMP in which only a single robot has a destination and the graph is planar, which we prove to be NP-complete. A crucial new ingredient for both of our results is that the solution admits a succinct representation as a small labeled topological minor of the input graph.

Figures (2)

• Figure 1: Left: an instance of CSMP with two robots with destinations (red and blue, with destinations marked as squares) and two blocker robots (both marked as solid black circles). Right: A representation of one particular optimal schedule for this instance, where we first move the two blockers and then the remaining two robots.
• Figure 2: An illustration for the reduction in Theorem \ref{['thm:nphardness']}. The left figure shows an instance of STR, i.e., a set $P$ of points, and a Steiner tree for $P$. The right figure shows the corresponding instance of Planar-CSMP-1.

Theorems & Definitions (36)

• Proposition A: Courcelle's Theorem Courcelle90ArnborgLS91
• Definition 1: Rooted graphs
• Definition 2: Topological minors of rooted graphs GroheKMW11
• Proposition 3
• Remark 4
• Definition B: Forest embedding and treedepth
• Proposition C: sparsity
• Definition 5
• Proposition E: Implied by YuR14
• Lemma F