Let P be a simple polygon, then the art gallery problem is looking for a minimum set of points (guards) that can see every point in P. We say two points a, b ∈ P can see each other if the line segment seg(a, b) is contained in P. We denote by V (P) the family of all minimum guard placements. The Hausdorff distance makes V (P) a metric space and thus a topological space. We show homotopy-universality, that is, for every semi-algebraic set S there is a polygon P such that V (P) is homotopy equivalent to S. Furthermore, for various concrete topological spaces T , we describe instances I of the art gallery problem such that V (I ) is homeomorphic to T .
We consider the algorithmic problem of finding the optimal weights and biases for a twolayer fully connected neural network to fit a given set of data points. This problem is known as empirical risk minimization in the machine learning community. We show that the problem is ∃R-complete. This complexity class can be defined as the set of algorithmic problems that are polynomial-time equivalent to finding real roots of a polynomial with integer coefficients. Our results hold even if the following restrictions are all added simultaneously.• There are exactly two output neurons.• There are exactly two input neurons.• The data has only 13 different labels.• The number of hidden neurons is a constant fraction of the number of data points.• The target training error is zero.• The ReLU activation function is used.
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