Complexity of Interlocking Polyominoes

TL;DR

This paper investigates the interlocking properties of polyominoes, proving that systems with five or fewer squares cannot interlock and that determining interlockedness with larger polyominoes is computationally hard (PSPACE hard).

Contribution

It establishes new bounds on interlocking capabilities of polyominoes and proves PSPACE hardness for systems including hexominoes or larger.

Findings

Polyominoes with five or fewer squares cannot interlock.

Determining interlockedness with hexominoes or larger is PSPACE hard.

Systems with four or fewer squares are non-interlocking.

Abstract

Polyominoes are a subset of polygons which can be constructed from integer-length squares fused at their edges. A system of polygons P is interlocked if no subset of the polygons in P can be removed arbitrarily far away from the rest. It is already known that polyominoes with four or fewer squares cannot interlock. It is also known that determining the interlockedness of polyominoes with an arbitrary number of squares is PSPACE hard. Here, we prove that a system of polyominoes with five or fewer squares cannot interlock, and that determining interlockedness of a system of polyominoes including hexominoes (polyominoes with six squares) or larger polyominoes is PSPACE hard.