The Minimum Vulnerability Problem |
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Authors: | Sepehr Assadi Ehsan Emamjomeh-Zadeh Ashkan Norouzi-Fard Sadra Yazdanbod Hamid Zarrabi-Zadeh |
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Affiliation: | 1. Department of Computer Engineering, Sharif University of Technology, Tehran, Iran
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Abstract: | 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 ({lfloor {k/2}rfloor }) -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(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 a 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. |
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