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

  • Authors:

  • Leonid Khachiyan;Endre Boros;Konrad Borys;Khaled Elbassioni;Vladimir Gurvich;Gabor Rudolf;Jihui Zhao

  • Affiliations:

  • 110 Frelinghuysen Road, 08854, NJ;Rutgers University, RUTCOR, 640 Bartholomew Road, 08854, Piscataway, NJ, USA;Rutgers University, RUTCOR, 640 Bartholomew Road, 08854, Piscataway, NJ, USA;Max-Planck-Institüt für Informatik, 640 Bartholomew Road, 08854, Saarbrücken, NJ, Germany;Rutgers University, RUTCOR, 640 Bartholomew Road, 08854, Piscataway, NJ, USA;Rutgers University, RUTCOR, 640 Bartholomew Road, 08854, Piscataway, NJ, USA;Rutgers University, Department of Computer Science, 110 Frelinghuysen Road, 08854, Piscataway, NJ, USA

  • Venue:
  • Theory of Computing Systems
  • Year:
  • 2008

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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 cs–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 $cd(s,t)≥d. Finally, we also show that the same inapproximability bounds hold for undirected graphs and/or node elimination.