Inapproximability of the Smallest Superpolyomino Problem

TL;DR

This paper proves that finding the smallest superpolyomino containing a set of colored polyominoes is computationally hard, with strong inapproximability results even for simple cases, highlighting fundamental complexity limits.

Contribution

It establishes NP-hardness of the smallest superpolyomino problem and demonstrates inapproximability bounds for multi-colored instances, advancing understanding of polyomino shape problems.

Findings

NP-hardness for single-color polyomino sets

NP-hardness to approximate within O(n^{1/3-ε}) for multi-color sets

Complexity results inform limitations of algorithmic solutions

Abstract

We consider the \emph{smallest superpolyomino problem}: given a set of colored polyominoes, find the smallest polyomino containing each input polyomino as a subshape. This problem is shown to be NP-hard, even when restricted to a set of polyominoes using a single common color. Moreover, for sets of polyominoes using two or more colors, the problem is shown to be NP-hard to approximate within a $O(n^{1/3-\varepsilon})$-factor for any $\varepsilon > 0$.