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Tiling layouts with dominoes

We explore the complexity of tiling finite subsets of the plane, which we call layouts, with a finite set of tiles. The tiles are inspired by Wang tiles and the domino game piece. Each tile is composed of a pair of faces. Each face is colored with one of possible colors. We want to know if a given layout is tileable by a given set of dominoes. In a tiling, dominoes that touch must do so at like-colored domino faces. We provide an time algorithm for tiling layouts that are paths or cycles. We also show that if the layout is partially tiled at the outset of the problem, then the tiling decision problem is NP-complete. We also show that the problem remains NP-complete even if the layout is a tree. In a geometric tiling problem we wish to fill all or some of the plane with non-overlapping polygons called tiles. The tiling problems studied herein are motivated by recent results concerning Wang tiles. Wang tiles are non-rotatable unit squares that have colored edges [5]. In a tiling that uses Wang tiles, neighboring tiles must have the same color on adjacent edges. In a typical Wang tiling problem, we are given a finite number of types of tiles and an infinite number of each type, and we are asked to tile some subset of the plane. Berger showed that deciding if the entire plane can be tiled by a given set of Wang tiles is undecidable [2]. Motivated by a connection between Wang tilings and self assembly in DNA computing, researchers have begun to study tiling proper infinite subsets of the plane [1, 3]. In [1], the authors show that the problem of tiling a ribbon, which is an infinite “path” in the plane, is undecidable. This result is extended in [3] to show that the problem of tiling a ribbon that is a “cycle” is undecidable. We study a variation of Wang tiles, which we call dominoes, that are rectangles that are partitioned into 2 colored faces. Thus unlike Wang tiles, the faces are colored rather than the edges. Also unlike Wang tiles, we allow rotation of the tiles and we consider finite sets of dominoes. Thus although our tiles have a connection to Wang tiles, they are essentially a generalization of the commonly used domino game piece.