Algorithms for translational tiling

Abstract

In this paper, we study algorithms for tiling problems. We show that the conditions (T1) and (T2) of Coven and Meyerowitz [E. Coven and A. Meyerowitz, Tiling the integers with translates of one finite set, J. Algebra 212(1) (1999), pp. 161–174], 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 ℤ144.