Classification and Traversal Algorithmic Techniques for Optimization Problems on Directed Hyperpaths

Abstract

Directed hypergraphs are used in several applications to modeldifferent combinatorial structures. A directed hypergraph is defined by a set of nodes and a set of hyperarcs, each connecting a set of source nodes to a single target node. A hyperpath, similarly to the notion of path in directed graphs, consists of a connection among nodes using hyperarcs. Unlike paths in graphs, however, hyperpaths are suitable of many different definitions of measure, which have been used in a wide set of applications. Not surprisingly, depending on theconsidered measure function the cost of finding optimal hyperpaths may range from NP-hard to linear time.  A first solution for finding optimal hyperpaths in case of a superior functions (SUP) can be found in a seminal work by Knuth [Knu77], which generalizes Dijkstra's Algorithm [Dij59] to deal with a grammar problem. This solution is further extended by Ramalingam and Reps [RR96] to deal with weakly superior functions (WSUP). Dijkstra's priority queue can find optimalpaths or hyperpaths if the measure function complies two hypotheses: it is mono- tone with respect to all its arguments and (multidimensional) triangle inequality holds. We show thatmonotonicity - alone - is sufficient to guarantee interestingproperties, and to make some optimization algorithms effective. Hence we introduce the generalized superior function (GSUP), and consider the symmetrical classes of inferior functions, giving rise to a hierarchy of classes of optimization problems on directed hypergraphs. After showing that some measure functions might induce cycles in optimal hyperpaths, we come up to another taxonomy of measure functions, based on the structure of the optimal hyperpaths they determine, and relate the two hierarchies.  Finally we introduce a general algorithmic pattern for the single-source optimal hyperpath problem encompassing existing and new algorithms, and compare their effectiveness in various cases, including the case of optimal cyclic hyperpaths.