A polycube in dimension d is a finite union of unit d-cubes whose vertices are on knots of the lattice Zd. We show that, for each family of polycubes E, there exists a finite set F of bricks (parallelepiped rectangles) such that the bricks which can be tiled by E are exactly the bricks which can be tiled by F. Consequently, if we know the set F, then we have an algorithm to decide in polynomial time if a brick is tilable or not by the tiles of E.
Marcus R. Garvie;John Burkardt, 2022, A Parallelizable Integer Linear Programming Approach for Tiling Finite Regions of the Plane with Polyominoes, Algorithms, 15, 5, pp. 164, 10.3390/a15050164, https://doi.org/10.3390/a15050164.
Serdar Karademir;Oleg A. Prokopyev;Robert J. Mailloux, 2015, Irregular polyomino tiling via integer programming with application in phased array antenna design, Journal of Global Optimization, 65, 2, pp. 137-173, 10.1007/s10898-015-0354-8.