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Extending Dijkstra’s Algorithm to Maximize the Shortest Path by Node-Wise Limited Arc Interdiction

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Computer Science – Theory and Applications (CSR 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3967))

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Abstract

We consider the problem of computing shortest paths in a directed arc-weighted graph G = (V,A) in the presence of an adversary that can block (interdict), for each vertex v ∈ V, a given number p(v) of the arcs A out (v) leaving v. We show that if all arc-weights are non-negative then the single-destination version of the problem can be solved by a natural extension of Dijkstra’s algorithm in time

$$O(|A|+|V|{\rm log}|V|+\Sigma_{\upsilon\in{V}\ \backslash \{t\}}(|A_{out}(\upsilon)|-p(\upsilon)){\rm log}(p(\upsilon)+1)).$$

Our result can be viewed as a polynomial algorithm for a special case of the network interdiction problem where the adversary’s budget is node-wise limited. When the adversary can block a given number p of arcs distributed arbitrarily in the graph, the problem (p-most-vital-arcs problem) becomes NP-hard. This result is also closely related to so-called cyclic games. No polynomial algorithm computing the value of a cyclic game is known, though this problem belongs to both NP and coNP.

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Author information

Authors and Affiliations

  1. Department of Computer Science, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey, 08854, USA

    Leonid Khachiyan & Jihui Zhao

  2. RUTCOR, Rutgers University, 640 Bartholomew Road, Piscataway, New Jersey, 08854, USA

    Vladimir Gurvich

Authors
  1. Leonid Khachiyan
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  2. Vladimir Gurvich
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  3. Jihui Zhao
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Editor information

Editors and Affiliations

  1. IRMAR, Université de Rennes, Campus de Beaulieu, 35042, Rennes Cedex, France

    Dima Grigoriev

  2. Intel Corporation, JF1-13, 2111 NE 25th Avenue, 97124, Hillsboro, OR, USA

    John Harrison

  3. Steklov Institute of Mathematics at St. Petersburg, 27 Fontanka, St., 191023, Petersburg, Russia

    Edward A. Hirsch

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© 2006 Springer-Verlag Berlin Heidelberg

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Khachiyan, L., Gurvich, V., Zhao, J. (2006). Extending Dijkstra’s Algorithm to Maximize the Shortest Path by Node-Wise Limited Arc Interdiction. In: Grigoriev, D., Harrison, J., Hirsch, E.A. (eds) Computer Science – Theory and Applications. CSR 2006. Lecture Notes in Computer Science, vol 3967. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11753728_24

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  • DOI: https://doi.org/10.1007/11753728_24

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-34166-6

  • Online ISBN: 978-3-540-34168-0

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