Parameterized approximation schemes for steiner trees with small number of Steiner vertices

We study the Steiner Tree problem, in which a set of terminal vertices needs to be connected in the cheapest possible way in an edge-weighted graph. This problem has been extensively studied from the viewpoint of approximation and also parametrization. In particular, on one hand Steiner Tree is known to be APX-hard, and W[2]-hard on the other, if parameterized by the number of non-terminals (Steiner vertices) in the optimum solution. In contrast to this we give an efficient parameterized approximation scheme (EPAS), which circumvents both hardness results. Moreover, our methods imply the existence of a polynomial size approximate kernelization scheme (PSAKS) for the considered parameter. We further study the parameterized approximability of other variants of Steiner Tree, such as Directed Steiner Tree and Steiner Forest. For neither of these an EPAS is likely to exist for the studied parameter: For Steiner Forest an easy observation shows that the problem is APX-hard, even if the input graph contains no Steiner vertices. For Directed Steiner Tree we prove that computing a constant approximation for this parameter is W[1]-hard. Nevertheless, we show that an EPAS exists for Unweighted Directed Steiner Tree. Also we prove that there is an EPAS and a PSAKS for Steiner Forest if in addition to the number of Steiner vertices, the number of connected components of an optimal solution is considered to be a parameter

Shell structure of atoms

The average local electrostatic potential function, V(r)/ρ(r), is calculated for 87 atoms and the corresponding monopositive and mononegative ions in the ground state, using the nonrelativistic average-over-configuration numerical Hartree-Fock density. It is found in general that the shell boundaries are expressed as the successively increasing maxima in V(r)/ρ(r) and the outermost maximum presents good approximate estimates of the core-valence separation in atoms. The reason for such behaviour is explained analytically. The proposed function is identified as the inverse of charge capacity per unit volume which attains a maximum at the shell boundary

Average local electrostatic potential and the core-valency separation in atoms

The average local electrostatic potential function, V(r)/ρ(r), is calculated for 87 atoms, Li-Ac, in the ground state using the nonrelativistic average-over-configuration numerical Hartree-Fock density. It is found empirically that in a given atom the shell boundaries are expressed as the successively increasing maxima in V(r)/ρ(r) and the outermost maximum presents good approximate estimates of the core-valence separation in atoms. The likeness in behavior of V(r)/ρ(r) at each shell boundary with the maximum hardness principle is discussed. The single-exponent-fit parameters for the electron density in the valency region are provided for all atoms