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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). [ABSTRACT FROM AUTHOR]

We show that two important problems that have applications in computational biology are ASP-complete, which implies that, given a solution to a problem, it is NP-complete to decide if another solution exists. We show first that a variation of Betweenness>, which is the underlying problem of questions related to radiation hybrid mapping, is ASP-complete. Subsequently, we use that result to show that Quartet Compatibility, a fundamental problem in phylogenetics that asks whether a set of quartets can be represented by a parent tree, is also ASP-complete. The latter result shows that Steel's Quartet Challenge, which asks whether a solution to Quartet Compatibility is unique, is coNP-complete. [ABSTRACT FROM PUBLISHER]