Computational Complexity of Motion Planning of a Robot through Simple Gadgets
This provides a foundational theory for understanding hardness in robot motion planning, with applications to video games like Zelda, though it is incremental in extending existing proof techniques.
The paper tackles the problem of analyzing the computational complexity of motion planning for a robot navigating through graphs of simple gadgets, showing that any nontrivial four-location two-state gadget makes motion planning PSPACE-complete, while simpler gadgets allow polynomial-time solutions.
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.