Polynomial-time decision procedure for polyomino cube-foldability

Establish whether there exists a polynomial-time algorithm that, given any abstract polyomino under the grid-crease folding model allowing only 90° and 180° folds along grid lines, decides whether the polyomino can be folded onto the surface of the unit cube (i.e., whether it is cube-foldable).

Background

The paper studies foldability of polyominoes into a unit cube under a simple folding model where creases are restricted to grid lines and fold angles of 90° or 180° are allowed. Prior work introduced the first correct general decision algorithm for this problem, but it runs in exponential time and uses unlink recognition from topology as a subroutine.

More tractable subclasses admit polynomial-time solutions (e.g., tree-shaped polyominoes). The overarching complexity question—whether cube-foldability can be decided in polynomial time in the general case—has been conjectured previously but remains unresolved; this paper contributes structural results for rectangular polyominoes with holes, yet does not resolve the general complexity question.