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The minimum vulnerability problem

Assadi, S ; Sharif University of Technology | 2012

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  1. Type of Document: Article
  2. DOI: 10.1007/s00453-014-9927-z
  3. Publisher: 2012
  4. Abstract:
  5. 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(n3/4), 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(√n). While the problem is NP-hard, and even hard to approximate to within an O(log n) 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
  6. Keywords:
  7. Approximation factor ; Input graphs ; K-paths ; NP-hard ; Planar graph ; Sparse graphs ; Sublinear ; Approximation algorithms ; Graph theory ; Problem solving
  8. Source: Algorithmica ; Volume 7676 LNCS , 2012 , Pages 382-391 ; 14320541(ISSN) ; 9783642352607 (ISBN)
  9. URL: http://link.springer.com/article/10.1007%2Fs00453-014-9927-z