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International Workshop on Graph-Theoretic Concepts in Computer Science

WG 2022: Graph-Theoretic Concepts in Computer Science pp 215–229Cite as

Generalized \(k\)-Center: Distinguishing Doubling and Highway Dimension

Part of the Lecture Notes in Computer Science book series (LNCS,volume 13453)

Abstract

We consider generalizations of the \(k\) -Center problem in graphs of low doubling and highway dimension. For the Capacitated \(k\) -Supplier with Outliers (CkSwO) problem, we show an efficient parameterized approximation scheme (EPAS) when the parameters are \(k\), the number of outliers and the doubling dimension of the supplier set. On the other hand, we show that for the Capacitated \(k\) -Center problem, which is a special case of CkSwO, obtaining a parameterized approximation scheme (PAS) is \(\mathrm {W[1]}\)-hard when the parameters are \(k\), and the highway dimension. This is the first known example of a problem for which it is hard to obtain a PAS for highway dimension, while simultaneously admitting an EPAS for doubling dimension.

Keywords

  • Capacitated \(k\)-Supplier with Outliers
  • Highway dimension
  • Doubling dimension
  • Parameterized approximation

Andreas Emil Feldmann was supported by the project 19-27871X of GA ČR. Tung Anh Vu was supported by the project 22-22997S of GA ČR.

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Fig. 1.
Fig. 2.

Notes

  1. 1.

    See [19, Section 9] for a discussion. In essence, the highway dimension of a given graph can vary depending on the selection of \(\gamma \).

  2. 2.

    See [13] or the full version of the paper for a formal definition.

  3. 3.

    We remark that for this distinction to work, one has to be careful of the used definition of highway dimension: a stricter definition of highway dimension from [1] already implies bounded doubling dimension. On the other hand, for certain types of transportation networks, it can be argued that the doubling dimension is large, while the highway dimension is small. See [22, Appendix A] for a detailed discussion.

References

  1. Abraham, I., Delling, D., Fiat, A., Goldberg, A.V., Werneck, R.F.: Highway dimension and provably efficient shortest path algorithms. J. ACM (JACM) 63(5), 1–26 (2016)

    CrossRef  MathSciNet  Google Scholar 

  2. Abraham, I., Fiat, A., Goldberg, A.V., Werneck, R.F.: Highway dimension, shortest paths, and provably efficient algorithms. In: Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms, pp. 782–793. SIAM (2010)

    Google Scholar 

  3. Ahmadi-Javid, A., Seyedi, P., Syam, S.S.: A survey of healthcare facility location. Comput. Oper. Res. 79, 223–263 (2017). https://doi.org/10.1016/j.cor.2016.05.018, https://www.sciencedirect.com/science/article/pii/S0305054816301253

  4. An, H.C., Bhaskara, A., Chekuri, C., Gupta, S., Madan, V., Svensson, O.: Centrality of trees for capacitated k-center. Math. Program. 154(1–2), 29–53 (2015). https://doi.org/10.1007/s10107-014-0857-y

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. Bast, H., Funke, S., Matijevic, D.: Transit ultrafast shortest-path queries with linear-time preprocessing. 9th DIMACS Implementation Challenge [1] (2006)

    Google Scholar 

  6. Bast, H., Funke, S., Matijevic, D., Sanders, P., Schultes, D.: In transit to constant time shortest-path queries in road networks. In: 2007 Proceedings of the Ninth Workshop on Algorithm Engineering and Experiments (ALENEX), pp. 46–59. SIAM (2007)

    Google Scholar 

  7. Becker, A., Klein, P.N., Saulpic, D.: Polynomial-time approximation schemes for k-center, k-median, and capacitated vehicle routing in bounded highway dimension. In: Azar, Y., Bast, H., Herman, G. (eds.) 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), vol. 112, pp. 8:1–8:15. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany (2018)

    Google Scholar 

  8. Blum, J.: Hierarchy of transportation network parameters and hardness results. In: Jansen, B.M.P., Telle, J.A. (eds.) 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), vol. 148, pp. 4:1–4:15. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany (2019)

    Google Scholar 

  9. Bodlaender, H.L., Lokshtanov, D., Penninkx, E.: Planar capacitated dominating set is W[1]-Hard. In: Chen, J., Fomin, F.V. (eds.) IWPEC 2009. LNCS, vol. 5917, pp. 50–60. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-11269-0_4

    CrossRef  Google Scholar 

  10. Chakrabarty, D., Goyal, P., Krishnaswamy, R.: The non-uniform k-center problem. ACM Trans. Algorithms 16(4) (2020). https://doi.org/10.1145/3392720

  11. Charikar, M., Khuller, S., Mount, D.M., Narasimhan, G.: Algorithms for facility location problems with outliers. In: SODA, vol. 1, pp. 642–651 (2001)

    Google Scholar 

  12. Cohen-Addad, V., Feldmann, A.E., Saulpic, D.: Near-linear time approximation schemes for clustering in doubling metrics. J. ACM 68(6), 1–34 (2021). https://doi.org/10.1145/3477541

    CrossRef  MathSciNet  Google Scholar 

  13. Cygan, M., et al.: Parameterized Algorithms. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-21275-3

    CrossRef  MATH  Google Scholar 

  14. Cygan, M., Hajiaghayi, M., Khuller, S.: LP rounding for k-centers with non-uniform hard capacities. In: 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, pp. 273–282. IEEE (2012)

    Google Scholar 

  15. Cygan, M., Kociumaka, T.: Constant factor approximation for capacitated k-center with outliers. In: Mayr, E.W., Portier, N. (eds.) 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), vol. 25, pp. 251–262. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany (2014). https://doi.org/10.4230/LIPIcs.STACS.2014.251, https://drops.dagstuhl.de/opus/volltexte/2014/4462

  16. Dom, M., Lokshtanov, D., Saurabh, S., Villanger, Y.: Capacitated domination and covering: a parameterized perspective. In: Grohe, M., Niedermeier, R. (eds.) IWPEC 2008. LNCS, vol. 5018, pp. 78–90. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-79723-4_9

    CrossRef  MATH  Google Scholar 

  17. Feder, T., Greene, D.: Optimal algorithms for approximate clustering. In: Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing, pp. 434–444. STOC 1988, Association for Computing Machinery, New York, NY, USA (1988). https://doi.org/10.1145/62212.62255,https://doi.org/10.1145/62212.62255

  18. Feldmann, A.E.: Fixed-parameter approximations for k-center problems in low highway dimension graphs. Algorithmica 81(3), 1031–1052 (2019). https://doi.org/10.1007/s00453-018-0455-0

    CrossRef  MathSciNet  MATH  Google Scholar 

  19. Feldmann, A.E., Fung, W.S., Konemann, J., Post, I.: A (1+\(\varepsilon \))-embedding of low highway dimension graphs into bounded treewidth graphs. SIAM J. Comput. 47(4), 1667–1704 (2018)

    CrossRef  MathSciNet  Google Scholar 

  20. Feldmann, A.E., Karthik, C., Lee, E., Manurangsi, P.: A survey on approximation in parameterized complexity: hardness and algorithms. Algorithms 13(6), 146 (2020)

    CrossRef  MathSciNet  Google Scholar 

  21. Feldmann, A.E., Marx, D.: The parameterized hardness of the k-center problem in transportation networks. Algorithmica 82(7), 1989–2005 (2020). https://doi.org/10.1007/s00453-020-00683-w

    CrossRef  MathSciNet  MATH  Google Scholar 

  22. Feldmann, A.E., Saulpic, D.: Polynomial time approximation schemes for clustering in low highway dimension graphs. J. Comput. Syst. Sci. 122, 72–93 (2021). https://doi.org/10.1016/j.jcss.2021.06.002, https://www.sciencedirect.com/science/article/pii/S0022000021000647

  23. Goyal, D., Jaiswal, R.: Tight FPT approximation for constrained k-center and k-supplier. CoRR abs/2110.14242 (2021). https://arxiv.org/abs/2110.14242

  24. Gupta, A., Krauthgamer, R., Lee, J.: Bounded geometries, fractals, and low-distortion embeddings. In: 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings, pp. 534–543. IEEE (2003)

    Google Scholar 

  25. Harris, D.G., Pensyl, T., Srinivasan, A., Trinh, K.: A lottery model for center-type problems with outliers. ACM Trans. Algorithms 15(3) (2019). https://doi.org/10.1145/3311953

  26. Hochbaum, D.S., Shmoys, D.B.: A best possible heuristic for the k-center problem. Math. Oper. Res. 10(2), 180–184 (1985)

    CrossRef  MathSciNet  Google Scholar 

  27. Hochbaum, D.S., Shmoys, D.B.: A unified approach to approximation algorithms for bottleneck problems. J. ACM 33(3), 533–550 (1986). https://doi.org/10.1145/5925.5933

    CrossRef  MathSciNet  Google Scholar 

  28. Katsikarelis, I., Lampis, M., Paschos, V.T.: Structural parameters, tight bounds, and approximation for (k, r)-center. Discret. Appl. Math. 264, 90–117 (2019)

    CrossRef  MathSciNet  Google Scholar 

  29. Talwar, K.: Bypassing the embedding: algorithms for low dimensional metrics. In: Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing, pp. 281–290. STOC 2004, Association for Computing Machinery, New York, NY, USA (2004). https://doi.org/10.1145/1007352.1007399

  30. Vazirani, V.V.: Approximation Algorithms. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-662-04565-7

    CrossRef  Google Scholar 

  31. Williamson, D.P., Shmoys, D.B.: The Design of Approximation Algorithmsd. Cambridge University Press, Cambridge (2011)

    CrossRef  Google Scholar 

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  1. Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

    Andreas Emil Feldmann & Tung Anh Vu

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  1. University of Ioannina, Ioannina, Greece

    Michael A. Bekos

  2. Universität Tübingen, Tübingen, Germany

    Michael Kaufmann

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Feldmann, A.E., Vu, T.A. (2022). Generalized \(k\)-Center: Distinguishing Doubling and Highway Dimension. In: Bekos, M.A., Kaufmann, M. (eds) Graph-Theoretic Concepts in Computer Science. WG 2022. Lecture Notes in Computer Science, vol 13453. Springer, Cham. https://doi.org/10.1007/978-3-031-15914-5_16

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