Minimum Exposure Motion Planning

TL;DR

This paper introduces a new 'Min-Exposure' objective for coordinated motion planning of robots, providing algorithms for minimal exposure schedules and proving fixed-parameter tractability for common objectives.

Contribution

It proposes the Min-Exposure objective, offers an algorithm for two robots, and proves FPT results for Min-Makespan and Min-Sum objectives.

Findings

Developed an $ ext{O}(n^4 ext{log} n)$ algorithm for 2 robots.

Established fixed-parameter tractability for Min-Makespan and Min-Sum objectives.

Extended FPT results to generalize known results for grid graphs.

Abstract

We investigate multiple fundamental variants of the classic coordinated motion planning (CMP) problem for unit square robots in the plane under the $L_1$ metric. In coordinated motion planning, we are given two arrangements of $k$ robots and are tasked with finding a movement schedule that minimizes a certain objective function. The two most prominent objective functions are the sum of distances traveled (Min-Sum) and the latest time of arrival (Min-Makespan). Both objectives have previously been studied extensively. We introduce a new objective function for CMP in the plane. The proposed Min-Exposure objective function defines a set of polygonal regions in the plane that provide cover and asks for a schedule with minimal elapsed time during which at least one robot is partially or fully outside of these regions. We give an $\mathcal{O}(n^4\log n)$ time algorithm that computes exposure-minimal schedules for $k=2$ robots, and an XP algorithm for arbitrary $k$. As a result of independent interest, we leverage new insights to prove that both the Min-Makespan and Min-Sum objectives are fixed-parameter tractable (FPT) parameterized by the number of robots. Our parameterized complexity results generalize known FPT results for rectangular grid graphs [Eiben, Ganian, and Kanj, SoCG'23].