A problem is defined to be exactly solvable if its solution can be obtained by solving a sequence of polynomials using radicals. Therefore, if a problem is not exactly solvable, then we have to use approximation schemes for solving the problem. It has been proved that the shortest network problem in space is not exactly solvable even if the network spans only four points and even if the topolog...

‎we describe a new type of presentation that‎, ‎when consistent‎, ‎describes a polycyclic group‎. ‎this presentation is obtained by‎ ‎refining a series of normal subgroups with‎ ‎abelian sections‎. ‎these presentations can be described effectively in‎ ‎computer-algebra-systems like {scshape gap} or ‎{scshape magma}‎. ‎we study these ‎presentations and‎, ‎in particular‎, ‎we obtain consistency c...