Parameterized Algorithms for Coordinated Motion Planning: Minimizing Energy

TL;DR

This paper develops parameterized algorithms for energy-efficient coordinated motion planning on graphs, extending results beyond grids, and establishes fixed-parameter tractability for certain graph classes while providing lower bounds for others.

Contribution

It introduces novel fixed-parameter algorithms for motion planning problems parameterized by robot count and graph properties, surpassing previous grid-restricted results.

Findings

Fixed-parameter additive approximation algorithm for energy minimization.

Fixed-parameter algorithm for single robot routing avoiding others.

Fixed-parameter algorithm for multi-robot routing on graphs of bounded treewidth.

Abstract

We study the parameterized complexity of a generalization of the coordinated motion planning problem on graphs, where the goal is to route a specified subset of a given set of $k$ robots to their destinations with the aim of minimizing the total energy (i.e., the total length traveled). We develop novel techniques to push beyond previously-established results that were restricted to solid grids. We design a fixed-parameter additive approximation algorithm for this problem parameterized by $k$ alone. This result, which is of independent interest, allows us to prove the following two results pertaining to well-studied coordinated motion planning problems: (1) A fixed-parameter algorithm, parameterized by $k$, for routing a single robot to its destination while avoiding the other robots, which is related to the famous Rush-Hour Puzzle; and (2) a fixed-parameter algorithm, parameterized by $k$ plus the treewidth of the input graph, for the standard \textsc{Coordinated Motion Planning} (CMP) problem in which we need to route all the $k$ robots to their destinations. The latter of these results implies, among others, the fixed-parameter tractability of CMP parameterized by $k$ on graphs of bounded outerplanarity, which include bounded-height subgrids. We complement the above results with a lower bound which rules out the fixed-parameter tractability for CMP when parameterized by the total energy. This contrasts the recently-obtained tractability of the problem on solid grids under the same parameterization. As our final result, we strengthen the aforementioned fixed-parameter tractability to hold not only on solid grids but all graphs of bounded local treewidth -- a class including, among others, all graphs of bounded genus.