## 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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## Notes

The runtime can be slightly improved to \(O({m ^ {2(k-1)}} {n ^ {3}})\) by discarding all cuts with no vulnerable edge in the cut sequences, and hence, by only considering cuts of size less that \(k\), as mentioned in the preliminary version of this work [1].

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## Acknowledgments

The authors would like to thank Jörg-Rüdiger Sack and Masoud T. Omran for their valuable discussions on the problem.

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This research is partially supported by IPM under Grant No. CS1391-4-04.

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Assadi, S., Emamjomeh-Zadeh, E., Norouzi-Fard, A. *et al.* The Minimum Vulnerability Problem.
*Algorithmica* **70**, 718–731 (2014). https://doi.org/10.1007/s00453-014-9927-z

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DOI: https://doi.org/10.1007/s00453-014-9927-z