Optimal shortest path set problem in undirected graphs

Abstract

This paper proposes the optimal shortest path set problem in an undirected graph $G=(V,E)$, in which some vehicles have to go from a source node $s$ to a destination node $t$. However, at most $k$ edges in the graph may be blocked during the traveling. The goal is to find a minimum collection of paths for the vehicles before they start off to assure the fastest arrival of at least one vehicle, no matter which $l$ $(0\le l\le k)$ edges are blocked. We consider two scenarios for this problem. In the first scenario with $k=1$, we propose the concept of common replacement path and design the Least-Overlap Algorithm to find the common replacement path. Based on this, an algorithm to compute the optimal shortest path set is presented and its time complexity is proved to be $O(n^2)$. In the second scenario with $k>1$, we consider the case where the blocked edges are consecutive ones on a shortest path from $s$ to $t$ and the vertices connecting two blocked edges are also blocked (i.e., routes passing through these vertices are not allowed), and an algorithm is presented to compute the optimal shortest path set in this scenario with time complexity $O(mn+k^2n^2\log n)$.