On the Parameterized Complexity of Motion Planning for Rectangular Robots

TL;DR

We address parameterized motion planning for $k$ axis-aligned rectangular robots under axis-aligned translations in both the free plane and a bounding box, with serial and parallel move models. The paper develops a grid-based structural lemma for serial axis-aligned motion in the free plane and several LP-backed, fixed-parameter algorithms when the set of translation directions is restricted, establishing FPT results parameterized by $k$ for most variants. It also delineates hardness boundaries, proving PSPACE-hardness for bounding-box variants and NP-hardness/NP-completeness for fixed-direction variants, thereby charting the tractability and intractability landscape of Rect-MP and Rect-CMP under axis-aligned constraints. Collectively, the work demonstrates that, despite geometric complexity, many natural axis-aligned motion-planning problems are fixed-parameter tractable in the number of robots, offering practical algorithms for instances with large input encodings.

Abstract

We study computationally-hard fundamental motion planning problems where the goal is to translate $k$ axis-aligned rectangular robots from their initial positions to their final positions without collision, and with the minimum number of translation moves. Our aim is to understand the interplay between the number of robots and the geometric complexity of the input instance measured by the input size, which is the number of bits needed to encode the coordinates of the rectangles' vertices. We focus on axis-aligned translations, and more generally, translations restricted to a given set of directions, and we study the two settings where the robots move in the free plane, and where they are confined to a bounding box. We obtain fixed-parameter tractable (FPT) algorithms parameterized by $k$ for all the settings under consideration. In the case where the robots move serially (i.e., one in each time step) and axis-aligned, we prove a structural result stating that every problem instance admits an optimal solution in which the moves are along a grid, whose size is a function of $k$, that can be defined based on the input instance. This structural result implies that the problem is fixed-parameter tractable parameterized by $k$. We also consider the case in which the robots move in parallel (i.e., multiple robots can move during the same time step), and which falls under the category of Coordinated Motion Planning problems. Finally, we show that, when the robots move in the free plane, the FPT results for the serial motion case carry over to the case where the translations are restricted to any given set of directions.

Figures (4)

• Figure 1: Top: robots at their starting position. Bottom: Robots (made smaller for visibility) can be separated by arbitrary large distance using a single direction. The numbers indicate the order in which the robots have to move. The figure is not at scale, as we might need very large distances to guarantee non-intersection in third and fourth moves of the robots.
• Figure 2: An illustration of a stacking to define new vertical lines.
• Figure 4: Illustration of the trace of a rectangle $abcd$ with respect to a vector $\overrightarrow{v}=\overrightarrow{oo'}$. Rectangle $a'b'c'd'=$translate($abcd, \overrightarrow{v})$ and the polygon $abcc'd'a'$, shown with solid lines, is trace($abcd, \overrightarrow{v}$). Observe that the edges of a trace are either edges of the rectangles, or are parallel to $\overrightarrow{v}$.
• Figure 5: Illustration of Subcases (C)-(2)-1 (left figure) and (C)-(2)-2 (right figure). In Subcase (C)-(2)-1, the distance $\Delta_V$ between the top edge of $R_q$ and the bottom edge of $R_p$ is larger than the distance $\Delta_H$ between the left edge of $R_p$ and the right edge of $R_q$. Hence, $R_p$ manages to "escape" $R_q$ in time. Similarly, in Subcase (C)-(2)-2, $R_q$ manages to escape $R_p$ in time.

Theorems & Definitions (15)

• Proposition 1
• Definition 2
• Proposition 3
• Proposition 4
• Definition 5
• Theorem 6
• Theorem 7
• Corollary 8
• Theorem 9
• Corollary 10