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 NPhard. 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.

Pseudopolynomial iterative algorithm to solve total-payoff games and min-cost reachability games

“…In the one-player setting (considering the point of view of Min for instance), this problem can be solved in polynomial time by Dijkstra's and Floyd-Warshall's algorithms when the weights are non-negative and arbitrary, respectively. Khachiyan et al [13] propose an extension of Dijkstra's algorithm to handle the two-player, non-negative weights case. However, in our more general setting (two players, arbitrary weights), this problem has, as far as we know, not been studied as such, except that the associated decision problem is known to be in NP ∩ co-NP [10].…”

Minimum d-blockers and d-transversals in graphs

“…It is known that a minimum cardinality s − t cut can be found in polynomial time (see [1]). The case d > 1 has been studied in [15] where the following observation is stated:…”

“…Proposition 5.1 [15] Let G be a graph such that evrey arc belongs to at least one shortest s − t path. T is an (inclusionwise) minimal d-transversal if and only if T is a union of d disjoint minimal s − t cuts.…”

“…arcs). In this context, a d-blocker is a subset B of arcs such that every s − t path in G = (V, E − B) has length at least l * (G) + d where l * (G) is the length of a shortest s − t path in G. Finding a minimum d-blocker in this case is known to be N P -hard (see [15]). …”

“…Such problems have been studied from a theoretical point of view and, in very special cases where a graph representation may be used, it leads to interesting combinatorial problems (see for instance [16], [15]). The purpose of this paper is to discuss some of these problems and to present a few variations with some remarks on their complexity and on open questions which would lead to further developments.…”