The Minimum Vulnerability Problem

  • Sepehr Assadi
  • Ehsan Emamjomeh-Zadeh
  • Ashkan Norouzi-Fard
  • Sadra Yazdanbod
  • Hamid Zarrabi-Zadeh
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7676)


We revisit the problem of finding k paths with a minimum number of shared edges between two vertices of a graph. An edge is called shared if it is used in more than one of the k paths. We provide a [k/2]-approximation algorithm for this problem, improving the best previous approximation factor of k − 1. We also provide the first approximation algorithm for the problem with a sublinear approximation factor of O(n 3/4), where n is the number of vertices in the input graph. For sparse graphs, such as bounded-degree and planar graphs, we show that the approximation factor of our algorithm can be improved to \(O(\sqrt{n})\). While the problem is NP-hard, and even hard to approximate to within an O(logn) factor, we show that the problem is polynomially solvable when k is a constant. This settles an open problem posed by Omran et al.  regarding the complexity of the problem for small values of k. We present most of our results in a more general form where each edge of the graph has a sharing cost and a sharing capacity, and there is vulnerability parameter r that determines the number of times an edge can be used among different paths before it is counted as a shared/vulnerable edge.


Approximation Algorithm Approximation Factor Network Design Problem Steiner Tree Problem Sparse Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Sepehr Assadi
    • 1
  • Ehsan Emamjomeh-Zadeh
    • 1
  • Ashkan Norouzi-Fard
    • 1
  • Sadra Yazdanbod
    • 1
  • Hamid Zarrabi-Zadeh
    • 1
    • 2
  1. 1.Department of Computer EngineeringSharif University of TechnologyTehranIran
  2. 2.Institute for Research in Fundamental Sciences (IPM)TehranIran

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