Theory of Computing Systems

, Volume 43, Issue 2, pp 204–233

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

• Leonid Khachiyan
• Endre Boros
• Khaled Elbassioni
• Gabor Rudolf
• Jihui Zhao
Open Access
Article

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,tV, 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 vV 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 st 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 $$c<10\sqrt{5}-21\approx1.36$$ 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.

Keywords

Approximation algorithm Dijkstra’s algorithm Most vital arcs problem Cyclic game Maxmin mean cycle Minimal vertex cover Network inhibition Network interdiction

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Authors and Affiliations

• Leonid Khachiyan
• Endre Boros
• 1
• 1
• Khaled Elbassioni
• 2