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Lately, it has been proposed to use shortest path first routing to implement Traffic Engineering in IP networks. The idea is to set the link weights so that the shortest paths, and the traffic thereof, follow the paths designated by the operator. Clearly, only certain shortest path representable path sets can be used in this setting, that is, paths which become shortest paths over some appropriately chosen positive, integer-valued link weights. Our main objective in this paper is to distill and unify the theory of shortest path representability under the umbrella of a novel flow-theoretic framework. In the first part of the paper, we introduce our framework and state a descriptive necessary and sufficient condition to characterize shortest path representable paths. Unfortunately, traditional methods to calculate the corresponding link weights usually produce a bunch of superfluous shortest paths, often leading to congestion along the unconsidered paths. Thus, the second part of the paper is devoted to reducing the number of paths in a representation to the bare minimum. To the best of our knowledge, this is the first time that an algorithm is proposed, which is not only able to find a minimal representation in polynomial time, but also assures link weight integrality. Moreover, we give a necessary and sufficient condition to the existence of a one-to-one mapping between a path set and its shortest path representation. However, as revealed by our simulation studies, this condition seems overly restrictive and instead, minimal representations prove much more beneficial.