OPTIMIZATION-BASED APPROACH TO TILING OF FINITE AREAS WITH ARBITRARY SETS OF WANG TILES

Wang tiles proved to be a convenient tool for the design of aperiodic tilings in computer graphics and in materials engineering. While there are several algorithms for generation of finite-sized tilings, they exploit the specific structure of individual tile sets, which prevents their general usage. In this contribution, we reformulate the NP-complete tiling generation problem as a binary linear program, together with its linear and semidefinite relaxations suitable for the branch and bound method. Finally, we assess the performance of the established formulations on generations of several aperiodic tilings reported in the literature, and conclude that the linear relaxation is better suited for the problem.