Complexity of Motion Planning of Arbitrarily Many Robots: Gadgets, Petri Nets, and Counter Machines

TL;DR

This paper analyzes the computational complexity of motion planning problems involving many robots, extending existing models to include spawning, destruction, and multi-player scenarios, revealing various complexity classifications including undecidability and EXPTIME-completeness.

Contribution

It introduces new models for multi-robot motion planning with spawning and destruction, and establishes their complexity classifications, including undecidability and high-level computational hardness.

Findings

Deciding if a robot reaches a location in the 0-player case is undecidable.

Reaching a location in the 1-player case is EXPSPACE-complete.

A shared-robot 2-player game with a ko rule is EXPTIME-complete.

Abstract

We extend the motion-planning-through-gadgets framework to several new scenarios involving various numbers of robots/agents, and analyze the complexity of the resulting motion-planning problems. While past work considers just one robot or one robot per player, most of our models allow for one or more locations to spawn new robots in each time step, leading to arbitrarily many robots. In the 0-player context, where all motion is deterministically forced, we prove that deciding whether any robot ever reaches a specified location is undecidable, by representing a counter machine. In the 1-player context, where the player can choose how to move the robots, we prove equivalence to Petri nets, EXPSPACE-completeness for reaching a specified location, PSPACE-completeness for reconfiguration, and ACKERMANN-completeness for reconfiguration when robots can be destroyed in addition to spawned. Finally, we consider a variation on the standard 2-player context where, instead of one robot per player, we have one robot shared by the players, along with a ko rule to prevent immediately undoing the previous move. We prove this impartial 2-player game EXPTIME-complete.