The next‐to‐shortest path problem on directed graphs with positive edge weights

Abstract

Given an edge-weighted graph G and two distinct vertices s and t of G, the next-to-shortest path problem asks for a path from s to t of minimum length among all paths from s to t except the shortest ones. In this article, we consider the version where G is directed and all edge weights are positive. Some properties of the requested path are derived when G is an arbitrary digraph. In addition, if G is planar, an O(n 3 )-time algorithm is proposed, where n is the number of vertices of G.

Simple paths with exact and forbidden lengths

“…Computational complexity is open if the graph contains directed cycles and the arc lengths are strictly positive. The latter case is solvable in O(n 3 ) time for planar graphs, as shown by Wu and Wang (2015). If the graph is undirected, then the problem is polynomially solvable.…”

“…If , then the shortest simple path is a solution of the instance of the problem P ath N o (). If , then the next‐to‐shortest simple path is a solution of this instance.Corollary The following special cases of the problem P ath N o ( ) are polynomially solvable : if graph is directed and planar and arc lengths are strictly positive, then P ath N o ( ) can be solved in time (Wu & Wang, ) ; if graph is undirected and arc lengths are strictly positive, then P ath N o ( ) can be solved in time (Kao et al, ) ; if graph is undirected and arc lengths are non‐negative, then P ath N o ( ) can be solved in time (Zhang & Nagamochi, ). Let us now show that for graphs with directed cycles and non‐negative arc lengths the problem P ath N o () is difficult. Observation If graph contains directed cycles and the arc lengths are all equal to 0 but one arc length is equal to 1, then any of the problems E xact P ath (), P ath N o (), P ath ‐1‐G ap , P ath ‐2‐G aps , P ath G aps , S hort P ath G aps and L ong P ath G aps is NP‐complete in the strong sense.Proof Fortune, Hopcroft, and Wyllie () proved that the problem T wo D isjoint P aths is NP‐complete in the strong sense.…”

“…• if graph G is directed and planar and arc lengths are strictly positive, then PATH NO(α) can be solved in O(n 3 ) time (Wu & Wang, 2015);…”