On Short Paths Interdiction Problems: Total and Node-Wise Limited Interdiction

Given a directed graph G =( V , A ) with a non-negative weight (length) function on its arcs w : A →ℝ + and two terminals s , t ∈ V , our goal is to destroy all short directed paths from s to t in G by eliminating some arcs of A . This is known as the short paths interdiction problem . We consider several versions of it, and in each case analyze two subcases: total limited interdiction , when a fixed number k of arcs can be removed, and node-wise limited interdiction , when for each node v ∈ V a fixed number k ( v ) of out-going arcs can be removed. Our results indicate that the latter subcase is always easier than the former one. In particular, we show that the short paths node-wise interdiction problem can be efficiently solved by an extension of Dijkstra’s algorithm. In contrast, the short paths total interdiction problem is known to be NP-hard. We strengthen this hardness result by deriving the following inapproximability bounds: Given k , it is NP-hard to approximate within a factor c <2 the maximum s – t distance d ( s , t ) obtainable by removing (at most) k arcs from G . Furthermore, given d , it is NP-hard to approximate within a factor c<10√(5)-21≈1.36 the minimum number of arcs which has to be removed to guarantee d ( s , t )≥ d . Finally, we also show that the same inapproximability bounds hold for undirected graphs and/or node elimination.