Abstract
Motivated by a security problem in geographic information systems, we study the following graph theoretical problem: given a graph G, two special nodes s and t in G, and a number k, find k paths from s to t in G so as to minimize the number of edges shared among the paths. This is a generalization of the well-known disjoint paths problem. While disjoint paths can be computed efficiently, we show that finding paths with minimum shared edges is NP-hard. Moreover, we show that it is even hard to approximate the minimum number of shared edges within a factor of \(2^{\log^{1-\varepsilon}n}\), for any constant ε>0. On the positive side, we show that there exists a (k−1)-approximation algorithm for the problem, using an adaption of a network flow algorithm. We design some heuristics to improve the quality of the output, and provide empirical results.
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Notes
The graph is available at: http://www.dis.uniroma1.it/~challenge9/data/rome/rome99.gr.
The generator is available at: https://sdm.lbl.gov/~kamesh/software/GTgraph/.
The information is available at: http://www.hpcvl.org/hpc-env-beowulf-cluster.html.
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Acknowledgements
The authors would like to thank Anil Maheshwari and Peter Widmayer for helpful discussions.
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Research supported by NSERC, SUN Microsystems and HPCVL.
A preliminary version of this work has been presented at COCOON 2011.
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Omran, M.T., Sack, JR. & Zarrabi-Zadeh, H. Finding paths with minimum shared edges. J Comb Optim 26, 709–722 (2013). https://doi.org/10.1007/s10878-012-9462-2
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DOI: https://doi.org/10.1007/s10878-012-9462-2