Shortest paths avoiding forbidden subpaths

Abstract

Abstract. In this paper we study a variant of the shortest path problem in graphs: given a weighted graph G and vertices s and t, and given a set X of forbidden paths in G, find a shortest s-t path P such that no path in X is a subpath of P . Path P is allowed to repeat vertices and edges. We call each path in X an exception, and our desired path a shortest exception avoiding path. We formulate a new version of the problem where the algorithm has no a priori knowledge of X, and finds out about an exception x ∈ X only when a path containing x fails. This situation arises in computing shortest paths in optical networks. We give an algorithm that finds a shortest exception avoiding path in time polynomial in |G| and |X|. The main idea is to run Dijkstra's algorithm incrementally after replicating vertices when an exception is discovered.

A RDBMS based Efficient Method for Shortest Path Searching over Large Graphs considering Constraint Conditions

“…Memory based shortest path searching methods considering constraint conditions [4][5] have been proposed. Ahmed et al [4] proposed an incremental algorithm for constructing shortest path searching tree that reflects constraint paths.…”

“…Ahmed et al [4] proposed an incremental algorithm for constructing shortest path searching tree that reflects constraint paths. Villeneuve et al [5] proposed a method to avoid the constraint path by pre-computing k-shortest paths.…”

Proper‐walk connection number of graphs

“…Walk‐restricted graphs can provide insight on structural properties of the underlying unrestricted graph (see, eg, [3] where antidirected walks are used to study 2‐detachments), or can be used to model practical situations. For example, forbidden‐transition graphs are used to solve routing problems in telecommunication networks [1] or in road networks [7] and edge‐coloured graphs are used in bioinformatics in [12]. In [19], Sudakov discusses how to measure the robustness of certain graph properties such as Hamiltonicity or connectivity and shows that it can sometimes be done by determining how many restrictions have to be put on the walks in a graph for the graph to lose the property.…”

On Minimum Connecting Transition Sets in Graphs

“…Depending on the applications, we sometimes need to express stronger constraints than what the standard definitions allow for. Indeed, in many practical cases, including optical networks, road networks or public transit systems among others, the set of possible walks a user can take is much more complex than the set of walks in a graph (see [1] or [2] for examples). To model a situation where a driver coming from a given road may not turn left while both the road he comes from and the road on the left exists, we have to define the permitted walks by taking into account not only the edges of the graph that a walk may use but also the transitions.…”