Covering a Polyomino-Shaped Stain with Non-Overlapping Identical Stickers

TL;DR

This paper classifies which polyomino-shaped stains can always be covered by identical non-overlapping stickers, provides an algorithm for this determination, and proves the problem's computational complexity as NP-complete.

Contribution

It offers a complete classification of always-coverable polyomino stains, constructs counterexamples, and establishes the NP-completeness of the coverability decision problem.

Findings

Maximal always-coverable polyominoes identified

Counterexamples for minimal not always-coverable polyominoes constructed

Coverability decision problem proven NP-complete

Abstract

You find a stain on the wall and decide to cover it with non-overlapping stickers of a single identical shape (rotation and reflection are allowed). Is it possible to find a sticker shape that fails to cover the stain? In this paper, we consider this problem under polyomino constraints and complete the classification of always-coverable stain shapes (polyominoes). We provide proofs for the maximal always-coverable polyominoes and construct concrete counterexamples for the minimal not always-coverable ones, demonstrating that such cases exist even among hole-free polyominoes. This classification consequently yields an algorithm to determine the always-coverability of any given stain. We also show that the problem of determining whether a given sticker can cover a given stain is $\NP$-complete, even though exact cover is not demanded. This result extends to the 1D case where the connectivity requirement is removed. As an illustration of the problem complexity, for a specific hexomino (6-cell) stain, the smallest sticker found in our search that avoids covering it has, although not proven minimum, a bounding box of $325 \times 325$.