Optimal Motion Planning for Two Square Robots in a Rectilinear Environment

TL;DR

This work addresses two-robot motion planning for axis-aligned unit-square robots in rectilinear environments, formalizing a 4D configuration space and two optimization goals: Min-Sum and Min-Makespan. It proves that the Min-Sum problem admits a polynomial-time exact algorithm by showing an optimal plan can be restricted to a canonical-grid on an $O(n)\\times O(n)$ grid, then embedding this grid into a 4D configuration graph of size $O(n^4)$ and solving via Dijkstra in $O(n^4\\log n)$. Conversely, it establishes NP-hardness for Min-Makespan through a partition-based reduction, highlighting a sharp complexity boundary. The approach hinges on structural concepts like influence regions, corridors, and a surgery framework that transforms any optimal plan into a canonical-grid plan without increasing cost, yielding the first polynomial-time exact algorithm for two translating unit-square robots in a planar polygonal environment. The results have theoretical significance for multi-robot planning and potential practical impact in domains requiring provable optimality guarantees for small robot teams in structured environments.

Abstract

Let $\mathcal{W} \subset \mathbb{R}^2$ be a rectilinear polygonal environment (that is, a rectilinear polygon potentially with holes) with a total of $n$ vertices, and let $A,B$ be two robots, each modeled as an axis-aligned unit square, that can move rectilinearly inside $\mathcal{W}$. The goal is to compute a collision-free motion plan $\boldsymbolπ$, that is, a motion plan that continuously moves $A$ from $s_A$ to $t_A$ and $B$ from $s_B$ to $t_B$ so that $A$ and $B$ remain inside $\mathcal{W}$ and do not collide with each other during the motion. We study two variants of this problem which are focused additionally on the optimality of $\boldsymbolπ$, and obtain the following results. 1. Min-Sum: Here the goal is to compute a motion plan that minimizes the sum of the lengths of the paths of the robots. We present an $O(n^4\log{n})$-time algorithm for computing an optimal solution to the min-sum problem. This is the first polynomial-time algorithm to compute an optimal, collision-free motion of two robots amid obstacles in a planar polygonal environment. 2. Min-Makespan: Here the robots can move with at most unit speed, and the goal is to compute a motion plan that minimizes the maximum time taken by a robot to reach its target location. We prove that the min-makespan variant is NP-hard.

Figures (22)

• Figure 1: The $0$-lines and $1$-lines define $\hbox{\sc hor}(\mathcal{F})$, which consist of twelve corridors. The dark gray region is an obstacle. To avoid cluttering the figure, the start and goal positions of the robots and the grid lines they define are omitted.
• Figure 2: An overview of the various regions inside $R_{\square}$ (not to scale). The regions $R_{\leftrightarrow}^-, R_{\leftrightarrow}, R_{\leftrightarrow}^+$ partition $R_{\square}$, and so do the regions $R_{\updownarrow}^-, R_{\updownarrow}, R_{\updownarrow}^+$. Corner squares are yellow. (Left) The height of $R$ is less than $1$, so the regions $Z^-, Z^+$ are disjoint from $R$. (Right) The height of $R$ is at least $1$, so the regions $Z^-, Z^+$ overlap $R$.
• Figure 3: An example influence region within $R_{\square}$ (not to scale), with the giant component $\gamma(R)$ drawn in green and the tiny components in the corner squares in yellow. In red, an illustration of a $pq$-path from Lemma \ref{['lem:xy-shortest path']}.
• Figure 4: Illustration of the proof of Lemma \ref{['lem:x-mono']}, using $\overline{\pi}$ and $\pi'$, where $h \subset \mathcal{F}$ is the horizontal segment between the black points. Lemma \ref{['lem:x-geodesic']} follows from setting $h = \mathrm{\textsf{top}}(R)$.
• Figure 5: (Left) An unsafe swap interval $[\nu_1,\nu_2] \subseteq [\lambda_1, \lambda_2]$ The robots become $y$-separated at time $\nu_1$ due to the vertical movement of $A$. At time $\nu_2$, robot $B$ leaves $\mathcal{I}(R)$. (Right) A swap interval, which is an unsafe interval with different $x$-orders at its endpoints.

Theorems & Definitions (81)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• Theorem 2.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 3.1
• proof