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Polynomial-Time Approximation Schemes for Subset-Connectivity Problems in Bounded-Genus Graphs

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Abstract

We present the first polynomial-time approximation schemes (PTASes) for the following subset-connectivity problems in edge-weighted graphs of bounded-genus: Steiner tree, low-connectivity survivable-network design, and subset TSP. The schemes run in \(\mathcal{O}(n \log n)\) time for graphs embedded on both orientable and nonorientable surfaces. This work generalizes the PTAS framework from planar graphs to bounded-genus graphs: any problem that is shown to be approximable by the planar PTAS framework of Borradaile et al. (ACM Trans. Algorithms 5(3), 2009) will also be approximable in bounded-genus graphs by our extension.

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Notes

  1. Before the publication of [14], we used [15] together with fast algorithms for finding shortest noncontractible cycles [10] but this can be avoided now.

  2. We would like to thank Christian Sommer for a discussion on this matter.

  3. Note that polynomial time is not crucial here as the number of portals is constant.

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Authors and Affiliations

  1. Oregon State University, 1148 Kelley Engineering Center, Corvallis, OR, 97331, USA

    Glencora Borradaile

  2. Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, 32 Vassar Streeet, Cambridge, MA, 02139, USA

    Erik D. Demaine & Siamak Tazari

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  1. Glencora Borradaile
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  2. Erik D. Demaine
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  3. Siamak Tazari
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Correspondence to Siamak Tazari.

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Borradaile, G., Demaine, E.D. & Tazari, S. Polynomial-Time Approximation Schemes for Subset-Connectivity Problems in Bounded-Genus Graphs. Algorithmica 68, 287–311 (2014). https://doi.org/10.1007/s00453-012-9662-2

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