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Superresolution imaging aims at improving the resolution of an image by enhancing it with other images or data that might have been acquired using different imaging techniques or modalities. In this paper we consider the task of doubling, in each dimension, the resolution of grayscale images of binary objects by fusion with double-resolution tomographic data that have been acquired from two viewing angles. We show that this task is polynomial-time solvable if the gray levels have been reliably determined. The problem becomes NP-hard if the gray levels of some pixels come with an error of ±1 or larger. The NP-hardness persists for any larger resolution enhancement factor. This means that noise does not only affect the quality of a reconstructed image but, less expectedly, also the algorithmic tractability of the inverse problem itself. [ABSTRACT FROM AUTHOR]

Hat problems have recently become a popular topic in combinatorics and discrete mathematics. These have been shown to be strongly related to coding theory, network coding, and auctions. We consider the following version of the hat game, introduced by Winkler and studied by Butler et al. A team is composed of several players; each player is assigned a hat of a given color; they do not see their own color but can see some other hats, according to a directed graph. The team wins if they have a strategy such that, for any possible assignment of colors to their hats, at least one player guesses their own hat color correctly. In this paper, we discover some new classes of graphs which allow a winning strategy, thus answering some of the open questions of Butler et al. We also derive upper bounds on the maximal number of possible hat colors that allow for a winning strategy for a given graph. [ABSTRACT FROM AUTHOR]

This is a paper about the beauty of the permutation method. New and shorter proofs are given for the theorem [P. L. Erdos and G. O. H. Katona, J. Combin. Theory. Ser. A, 43 (1986), pp. 58--69; S. Shahriari, Discrete Math., 162 (1996), pp. 229--238] determining all extremal two-part Sperner families and for the uniqueness of k-Sperner families of maximum size [P. Erdos, Bull. Amer. Math. Soc., 51 (1945), pp. 898--902]. [ABSTRACT FROM AUTHOR]