Minimum Number Of Faces Research Articles

Asymptotic properties of the estimate refinement method in polyhedral approximation of multidimensional balls

The estimate refinement method for the polyhedral approximation of convex compact bodies is considered. In the approximation of convex bodies with a smooth boundary, this method is known to generate polytopes with an optimal order of growth of the number of vertices and facets depending on the approximation error. The properties of the method are examined as applied to the polyhedral approximation of a multidimensional ball. As vertices of approximating polytopes, the method is shown to generate a deep holes sequence on the surface of the ball. As a result, previously obtained combinatorial properties of convex hulls of the indicated sequences, namely, the convergence rates with respect to the number of faces of all dimensions and the optimal growth of the cardinality of the facial structure (of the norm of the f-vector) can be extended to such polytopes. The combinatorial properties of the approximating polytopes generated by the estimate refinement method are compared to the properties of polytopes with a facial structure of extremal cardinality. It is shown that the polytopes generated by the method are similar to stacked polytopes, on which the minimum number of faces of all dimensions is attained for a given number of vertices.

Optimizing Phylogenetic Networks for Circular Split Systems

We address the problem of realizing a given distance matrix by a planar phylogenetic network with a minimum number of faces. With the help of the popular software SplitsTree4, we start by approximating the distance matrix with a distance metric that is a linear combination of circular splits. The main results of this paper are the necessary and sufficient conditions for the existence of a network with a single face. We show how such a network can be constructed, and we present a heuristic for constructing a network with few faces using the first algorithm as the base case. Experimental results on biological data show that this heuristic algorithm can produce phylogenetic networks with far fewer faces than the ones computed by SplitsTree4, without affecting the approximation of the distance matrix.