The Shortest Path Problem with Edge Information Reuse is NP-Complete

We show that the following variation of the single-source shortest path problem is NP-complete. Let a weighted, directed, acyclic graph G = (V, E, w) with source and sink vertices s and t be given. Let in addition a mapping f on E be given that associate information with the edges (e.g., a pointer), such that f (e) = f (e ′) means that edges e and e ′ carry the same information; for such edges it is required that w(e) = w(e ′). The length of a simple st path U is the sum of the weights of the edges on U but edges with f (e) = f (e ′) are counted only once. The problem is to determine a shortest such st path. We call this problem the edge information reuse shortest path problem. It is NP-complete by reduction from PARTITION. A weighted, directed, acyclic graph G = (V, E, w) with source and sink vertices s, t ∈ V is given. Edges represent some possible substructures and a path from s to t determines how substructures are put together to form a desired superstructure. Edge weights reflect the cost of the associated substructures (e.g., memory consumption). An ordinary shortest path determines an ordered, tree-like representation of the superstructure of least cost: a single root with children given by the edges of the path. Now, different edges in G may represent similar substructures, and if several such edges occur on a path, the cost of the substructure need be paid only once. We want to find a shortest such path from s to t where the weight of edges with similar information is counted only once. There may easily be such a path of less cost than an ordinary shortest path in G. In that case, a more cost-efficient representation of the desired superstructure is by a compressed tree where several children are represented by the same substructure. We model this problem by associating information with the edges in the form of a function f on E that may for instance be a pointer to the substructure represented by the edge. Two edges e, e ′ ∈ E carry the same information and are similar in that respect if f (e) = f (e ′). For the optimization problem to be well-defined, it is required that edges with f (e) = f (e ′) have the same …