Computational Complexity of Motion Planning of a Robot through Simple Gadgets

TL;DR

This paper develops a general framework for analyzing the computational complexity of robot motion planning through gadgets, showing that certain configurations lead to PSPACE-complete problems, while simpler ones are polynomial.

Contribution

It characterizes the complexity of robot motion planning with gadgets, identifying conditions under which the problem is PSPACE-complete versus polynomial-time solvable.

Findings

Any single nontrivial four-location two-state gadget makes planning PSPACE-complete.

Simpler gadgets with fewer locations or states have polynomial-time algorithms.

Motion planning games with spinners are PSPACE-complete, like in Zelda: Oracle of Seasons.

Abstract

We initiate a general theory for analyzing the complexity of motion planning of a single robot through a graph of "gadgets", each with their own state, set of locations, and allowed traversals between locations that can depend on and change the state. This type of setup is common to many robot motion planning hardness proofs. We characterize the complexity for a natural simple case: each gadget connects up to four locations in a perfect matching (but each direction can be traversable or not in the current state), has one or two states, every gadget traversal is immediately undoable, and that gadget locations are connected by an always-traversable forest, possibly restricted to avoid crossings in the plane. Specifically, we show that any single nontrivial four-location two-state gadget type is enough for motion planning to become PSPACE-complete, while any set of simpler gadgets (effectively two-location or one-state) has a polynomial-time motion planning algorithm. As a sample application, our results show that motion planning games with "spinners" are PSPACE-complete, establishing a new hard aspect of Zelda: Oracle of Seasons.