Abstract
We give an exact \(O(nk^2)\) algorithm for finding the densest k subgraph in outerplanar graphs. We extend this to an exact \(O(nk^2 8^b)\) algorithm for finding the densest k subgraph in b-outerplanar graphs. Often, when there is an exact polynomial time algorithm for a problem on b-outerplanar graphs, this algorithm can be extended to a polynomial time approximation scheme (PTAS) on planar graphs using Baker’s technique. We hypothesize that this is not possible for the densest k subgraph problem.
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Acknowledgements
We would like to express our sincere thanks to Samuel Chase for his collaboration on our initial explorations of finding a PTAS for the densest k subgraph problem. We would like to thank our reviewer who pointed us to previous work on this problem [3].
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Gonzales, S., Migler, T. (2020). The Densest k Subgraph Problem in b-Outerplanar Graphs. In: Cherifi, H., Gaito, S., Mendes, J., Moro, E., Rocha, L. (eds) Complex Networks and Their Applications VIII. COMPLEX NETWORKS 2019. Studies in Computational Intelligence, vol 881. Springer, Cham. https://doi.org/10.1007/978-3-030-36687-2_10
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DOI: https://doi.org/10.1007/978-3-030-36687-2_10
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