Structure theorems for optimum hyperpaths in directed hypergraphs

The cost of hyperpaths in directed hypergraphs can be measuread in various different ways, which have been used in a wide set of applications. Not surprisingly, depending on the considered measure function the cost to find optimum hyperpaths may range from NP-hard to linear time. A first solution for finding optimum hyperpaths in case of a superior functions (SUP) can be found in a seminal work by Knuth [5], which generalizes Dijkstra’s Algorithm [3] to deal with a grammar problem. In this paper we define a hierarchy of classes of optimization problems based on the properties of the cost measures. After showing that measures can be classified on the basis of the structure of the optimum hyperpath they determine, we present an alternative taxonomy of measure functions, based on their analytic properties, and prove structure theorems that relate the two hierarchies.