Parameterized Algorithms for Coordinated Motion Planning: Minimizing Energy

TL;DR

This work advances the parameterized complexity of Generalized Coordinated Motion Planning (GCMP) on general graphs with an energy-minimization objective. It introduces an additive FPT approximation for GCMP parameterized by the number of robots $k$, enabling two main results: (i) GCMP1 is fixed-parameter tractable when parameterized by $k$, and (ii) CMP is fixed-parameter tractable when parameterized by $(k,\text{treewidth})$, yielding tractability on graph classes with bounded outerplanarity. It also establishes a hardness barrier: CMP parameterized by the total energy $\ell$ is W[1]-hard on general graphs, though GCMP is FPT by $\ell$ on graphs with bounded local treewidth. The authors further extend FPT results to treewidth-bounded graphs via a tree-decomposition based dynamic programming framework that uses boundaried graphs, signatures, and checkpoints, and they generalize beyond solid grids to broader graph families. Overall, the paper delivers a cohesive set of algorithms and lower bounds that substantially broaden the landscape of tractable multirobot motion planning problems.

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.

Figures (1)

• Figure 1: Illustration of the four types for a vertex $v$ that is not nice.

Theorems & Definitions (38)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Definition 6: Tree decomposition
• Definition 7: Nice tree decomposition
• Proposition A: Korhonen21
• Definition B: Boundaried Graph
• Definition C