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

Abstract

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 non-directed graphs and/or node elimination.Keywords: approximation algorithm, Dijkstra's algorithm, most vital arcs problem, cyclic game, maxmin mean cycle, minimal vertex cover, network inhibition, network interdiction. Introduction Node-wise limited interdictionLet G = (V, A) be a directed graph (digraph) with given arc-weights w(e), e ∈ A. For each vertex v ∈ V , we are allowed to delete (remove, block, interdict) a subset X(v) of the arcs A(v) = {e ∈ A | e = (v, u)} leaving v. We assume that these arc-sets X(v) ⊆ A(v) are selected for all vertices v ∈ V independently, and we call the collection B(v) of all admissible arc-sets X(v) a blocking system at v. We also assume that for each v, the family B(v) forms an independence system, i.e., if X(v) ∈ B(v) is an admissible arc-set at v, then so is any subset of X(v). Hence, we could replace B(v) by the collection of all inclusion-wise maximal admissible arc-sets. In general, we will only assume that the blocking system B(v) is given by a membership oracle O : (B 0 ) Given a list X(v) of out-going arcs for some vertex v, the oracle can determine whether or not the arcs in the list belong to B(v) and hence can be simultaneously deleted.A similar formalization of blocking sets via membership oracles is used by Pisaruk [37]. We will also consider two special types of blocking systems:For each vertex v, we can delete any collection of (at most) k(v) arcs leaving v. The numbers k(v) define digraphs with prohibitions considered by Karzanov and Lebedev [30].(B 2 ) There are two types of vertices: control vertices, where we can select any out-going arc e ∈ A(v) and block all the remaining arcs in A(v), and regular vertices, where we can block no arc. This case, considered...

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