[Paper Review] Toward a General Theory of Motion Planning Complexity: Characterizing Which Gadgets Make Games Hard

Erik D. Demaine, Dylan Hendrickson+1 more|arXiv (Cornell University)|Dec 10, 2018

Logic, Reasoning, and Knowledge4 citations

TL;DR

This paper develops a general complexity theory for motion planning in games and robotics by analyzing how specific 'gadgets'—state-changing components that control robot traversal—affect computational hardness. It fully characterizes the complexity of 1-player and 2-player motion planning problems based on gadget types, showing that certain gadgets (e.g., locking 2-toggles, one-way tunnels) suffice to make problems NP-complete, PSPACE-complete, or even undecidable, offering a unified framework to simplify and generalize prior hardness proofs.

ABSTRACT

We build a general theory for characterizing the computational complexity of motion planning of robot(s) through a graph of "gadgets", where each gadget has its own state defining a set of allowed traversals which in turn modify the gadget's state. We study two families of such gadgets, one which naturally leads to motion planning problems with polynomially bounded solutions, and another which leads to polynomially unbounded (potentially exponential) solutions. We also study a range of competitive game-theoretic scenarios, from one player controlling one robot to teams of players each controlling their own robot and racing to achieve their team's goal. Under small restrictions on these gadgets, we fully characterize the complexity of bounded 1-player motion planning (NL vs. NP-complete), unbounded 1-player motion planning (NL vs. PSPACE-complete), and bounded 2-player motion planning (P vs. PSPACE-complete), and we partially characterize the complexity of unbounded 2-player motion planning (P vs. EXPTIME-complete), bounded 2-team motion planning (P vs. NEXPTIME-complete), and unbounded 2-team motion planning (P vs. undecidable). These results can be seen as an alternative to Constraint Logic (which has already proved useful as a basis for hardness reductions), providing a wide variety of agent-based gadgets, any one of which suffices to prove a problem hard.

Motivation & Objective

• To develop a general theory that characterizes which gadgets make motion planning problems computationally hard.
• To classify the complexity of 1-player and 2-player motion planning based on gadget types, including bounded and unbounded move counts.
• To provide a framework that generalizes and simplifies existing hardness proofs in games and robotics.
• To identify minimal gadget sets that suffice to prove hardness across various game types and complexity classes.
• To explore whether planar systems of gadgets yield different complexity than non-planar ones.

Proposed method

• Model motion planning as robot traversal through a graph of stateful gadgets, where each traversal modifies the gadget’s state and future traversal rules.
• Define two main gadget families: DAG gadgets (polynomially bounded solutions) and reversible deterministic gadgets (potentially exponential solutions).
• Use reductions from known hard problems (e.g., 3SAT, QSAT, planar motion planning) to prove hardness for specific gadget configurations.
• Construct concrete gadget realizations in existing games (e.g., PushPull-1F, Mario Kart) to demonstrate applicability and simplify proofs.
• Apply the framework to prove complexity results for competitive, team-based, and single-agent motion planning under various move bounds.
• Provide full or partial characterizations of complexity classes (NL, NP, PSPACE, EXPTIME, NEXPTIME, undecidable) based on gadget properties.

Experimental results

Research questions

• RQ1 Which gadget types are sufficient to make 1-player motion planning NP-complete or PSPACE-complete?
• RQ2 Can the complexity of 2-player and team-based motion planning be fully characterized based on gadget types?
• RQ3 Do reversible deterministic gadgets lead to exponential solution lengths, and how does this affect complexity?
• RQ4 Can planar configurations of gadgets yield different complexity than non-planar ones?
• RQ5 Can this framework simplify or replace lengthy prior hardness proofs in games like Lemmings or Mario Kart?

Key findings

• 1-player motion planning with DAG gadgets is either in NL or NP-complete, depending on whether the gadget set allows for a polynomially bounded solution.
• Unbounded 1-player motion planning is either in NL or PSPACE-complete, with the latter arising when gadgets allow for exponential-length solution paths.
• Bounded 2-player motion planning is either in P or PSPACE-complete, depending on the presence of certain reversible or one-way gadgets.
• Unbounded 2-player motion planning is either in P or EXPTIME-complete, with the latter class achievable via specific gadget configurations.
• Bounded 2-team motion planning is either in P or NEXPTIME-complete, and unbounded 2-team motion planning can be undecidable under certain gadget sets.
• The framework simplifies prior hardness proofs: for example, the PSPACE-hardness of Mario Kart’s 2-player mode now follows from a single, simple gadget construction, reducing a multi-page proof to a paragraph.