[Paper Review] Motion planning with pull moves

Marcus Ritt|arXiv (Cornell University)|Aug 17, 2010

Artificial Intelligence in Games10 references10 citations

TL;DR

This paper proves the NP-hardness of pull-only variants of Sokoban, where the warehouse keeper can only pull boxes instead of pushing them. Using a reduction from planar 3-colorability, the author constructs specialized gadgets (one-way, fork, NAND, and XOR) that function under pull mechanics, demonstrating that even with limited movement capabilities, the motion planning problem remains computationally intractable.

ABSTRACT

It is well known that Sokoban is PSPACE-complete (Culberson 1998) and several of its variants are NP-hard (Demaine et al. 2003). In this paper we prove the NP-hardness of some variants of Sokoban where the warehouse keeper can only pull boxes.

Motivation & Objective

• To investigate the computational complexity of motion planning in Sokoban variants where the robot can only pull boxes.
• To determine whether pull-only mechanics still yield NP-hard problems, despite the restriction on movement direction.
• To extend prior hardness results from push-based variants to pull-based counterparts using a novel reduction strategy.
• To construct and validate pull-specific implementations of core gadgets (e.g., one-way, fork, NAND) used in NP-hardness proofs.
• To explore the feasibility of isolating gadgets and preventing unintended path openings when multiple boxes are pulled simultaneously.

Proposed method

• Reduction from planar 3-colorability to construct a pull-based Sokoban puzzle instance.
• Design of pull-specific implementations for elementary gadgets: one-way, 1-to-3 fork, XOR-crossing, and NAND-gadgets.
• Verification that pull mechanics do not open new traversal paths compared to push-based versions.
• Use of thickened walls and spatial constraints to prevent escape or unintended movement in movable-box configurations.
• Simulation of all possible box movements to confirm that only intended paths remain valid under pull rules.
• Extension of the proof to Pull-k-F, PullPull-k-F, and Pull-1 variants by showing that increased pulling capacity does not reduce problem hardness.

Experimental results

Research questions

• RQ1 Is the motion planning problem in Sokoban NP-hard when the robot is restricted to pulling boxes instead of pushing them?
• RQ2 Can the standard NP-hardness proof techniques for push-based Sokoban be adapted to pull-only mechanics?
• RQ3 Do pull-based gadgets (e.g., one-way, fork, NAND) behave equivalently to their push-based counterparts in terms of path control and choice enforcement?
• RQ4 Does allowing the simultaneous pulling of multiple boxes (k > 1) or an unlimited number (k = *) reduce the problem's computational hardness?
• RQ5 Can pull-only variants of Sokoban be shown to be PSPACE-complete, or is NP-hardness the tightest known bound?

Key findings

• Pull-1-F is NP-hard, proven via a reduction from planar 3-colorability using pull-specific gadget implementations.
• All Pull-k-F variants are NP-hard for any k ∈ ℕ ∪ {*}, as increased pulling capacity does not enable new paths or simplify the puzzle.
• PullPull-k-F variants are also NP-hard for any k ∈ ℕ, as the pull-only movement constraints preserve the structural integrity of the gadgets.
• The Pull-1 variant (without fixed obstacles) is NP-hard, as movable boxes do not open unintended paths and the robot remains confined within gadget structures.
• The proof strategy simplifies prior push-based hardness proofs by introducing a new branch gadget and avoiding complex interference issues.
• The authors conjecture that pull-based variants remain NP-hard even under stronger constraints, such as directional handles or restricted pulling directions.