Algorithms for translational tiling

TL;DR

This paper develops polynomial-time algorithms to verify Coven-Meyerowitz conditions for tiling integers and introduces heuristic methods for classifying non-periodic tilings of cyclic groups, including a complete classification of Z_144.

Contribution

It provides the first polynomial-time algorithms for checking tiling conditions and offers heuristic algorithms for classifying non-periodic tilings, with a full classification of Z_144.

Findings

Conditions (T1) and (T2) can be checked in polynomial time.

Heuristic algorithms effectively find non-periodic tilings.

Complete classification of non-periodic tilings of Z_144.

Abstract

In this paper we study algorithms for tiling problems. We show that the conditions $(T1)$ and $(T2)$ of Coven and Meyerowitz, conjectured to be necessary and sufficient for a finite set $A$ to tile the integers, can be checked in time polynomial in ${diam}(A)$. We also give heuristic algorithms to find all non-periodic tilings of a cyclic group $Z_N$. In particular we carry out a full classification of all non-periodic tilings of $Z_{144}$.