Computing Internet routing metrics

In spite of the development of Internet networks and the important volume of literature dealing with Internet routing, many fundamental topics were not addressed. Traffic is, in principle, routed through shortest paths, in sense of a set of link weights (a metric), These weights do not necessarily have a physical significance and could be modified by the network administrator to change the routing policy and the network cost. In this paper, we give precise answers for the following questions: 1/ Given a network topology, is there a metric such that, first, the shortest path between any pair of vertices is unique and, second, every link belongs to at least one shortest path? 2/ How can we compute such a metric? 3/ Can we choose a metric satisfying these constraints and whose values are small integers? 4/ If the routers of the network have differents functions and characteristics, is it possible to determine a metric which allows to route traffic, taking into account these heterogeneous characteristics? 5/ If some routing paths are selected due to technical or economical reasons, can we find a metric enforcing this given routing policy? 6/ If this is not possible, what should we do to compute a metric that is as close as possible to the selected routing paths?