Search

We study the computational hardness of the tiling puzzle with polyominoes, where a polyomino is a right-angled polygon (i.e., a polygon made by connecting unit squares along their edges). In the tiling problem, we are given a right-angled polygon P and a set S of polyominoes, and asked whether P can be covered without any overlap using translated copies of polyominoes in S. In this paper, we focus on trominoes and bars as polyominoes; a tromino is a polyomino consisting of three unit squares, and a bar is a rectangle of either height one or width one. Notice that there are essentially two shapes of trominoes, that is, I-shape (i.e., a bar) and L-shape. We consider the tiling problem when restricted to only L-shape trominoes, only I-shape trominoes, both L-shape and I-shape trominoes, or only two bars. In this paper, we prove that the tiling problem remains NP-complete even for such restricted sets of polyominoes. All reductions are carefully designed so that we can also prove the # P-completeness and ASP-completeness of the counting and the another-solution-problem variants, respectively. Our results answer two open questions proposed by Moore and Robson (Discrete Comput Geom 26:573–590, 2001) and Pak and Yang (J Comb Theory 120:1804–1816, 2013).

Interval temporal logics are based on temporal structures where time intervals, rather than time instants, are the primitive ontological entities. They employ modal operators corresponding to various relations between intervals, known as Allen’s relations. Technically, validity in interval temporal logics translates to dyadic second-order logic, thus explaining their complex computational behavior. The full modal logic of Allen’s relations, called HS, has been proved to be undecidable by Halpern and Shoham under very weak assumptions on the class of interval structures, and this result was discouraging attempts for practical applications and further research in the field. A renewed interest has been recently stimulated by the discovery of interesting decidable fragments of HS. This paper contributes to the characterization of the boundary between decidability and undecidability of HS fragments. It summarizes known positive and negative results, it describes the main techniques applied so far in both directions, and it establishes a number of new undecidability results for relatively small fragments of HS.