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Bicriteria Approximation for Minimum Dilation Graph Augmentation

Authors Kevin Buchin , Maike Buchin , Joachim Gudmundsson , Sampson Wong

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  • Theory of computation → Computational geometry
Keywords
  • Greedy spanner
  • Graph augmentation

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Abstract

Spanner constructions focus on the initial design of the network. However, networks tend to improve over time. In this paper, we focus on the improvement step. Given a graph and a budget k, which k edges do we add to the graph to minimise its dilation? Gudmundsson and Wong [TALG'22] provided the first positive result for this problem, but their approximation factor is linear in k. 
Our main result is a (2 √[r]{2} k^{1/r},2r)-bicriteria approximation that runs in O(n³ log n) time, for all r ≥ 1. In other words, if t^* is the minimum dilation after adding any k edges to a graph, then our algorithm adds O(k^{1+1/r}) edges to the graph to obtain a dilation of 2rt^*. Moreover, our analysis of the algorithm is tight under the Erdős girth conjecture.

Cite As Get BibTex

Kevin Buchin, Maike Buchin, Joachim Gudmundsson, and Sampson Wong. Bicriteria Approximation for Minimum Dilation Graph Augmentation. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 36:1-36:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.36

Author Details

Kevin Buchin
  • Technical University Dortmund, Germany
Maike Buchin
  • Ruhr University Bochum, Germany
Joachim Gudmundsson
  • University of Sydney, Australia
Sampson Wong
  • University of Copenhagen, Denmark

References

  1. Abu Reyan Ahmed, Greg Bodwin, Faryad Darabi Sahneh, Keaton Hamm, Mohammad Javad Latifi Jebelli, Stephen G. Kobourov, and Richard Spence. Graph spanners: A tutorial review. Computer Science Review, 37:100253, 2020. URL: https://doi.org/10.1016/j.cosrev.2020.100253.
  2. Ingo Althöfer, Gautam Das, David P. Dobkin, Deborah Joseph, and José Soares. On sparse spanners of weighted graphs. Discrete & Computational Geometry, 9:81-100, 1993. URL: https://doi.org/10.1007/BF02189308.
  3. Boris Aronov, Kevin Buchin, Maike Buchin, Bart M. P. Jansen, Tom de Jong, Marc J. van Kreveld, Maarten Löffler, Jun Luo, Rodrigo I. Silveira, and Bettina Speckmann. Connect the dot: Computing feed-links for network extension. Journal of Spatial Information Science, 3(1):3-31, 2011. URL: https://doi.org/10.5311/JOSIS.2011.3.47.
  4. Boris Aronov, Mark de Berg, Otfried Cheong, Joachim Gudmundsson, Herman J. Haverkort, Michiel H. M. Smid, and Antoine Vigneron. Sparse geometric graphs with small dilation. Computational Geometry, 40(3):207-219, 2008. URL: https://doi.org/10.1016/j.comgeo.2007.07.004.
  5. Mihai Badoiu, Piotr Indyk, and Anastasios Sidiropoulos. Approximation algorithms for embedding general metrics into trees. In Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007. URL: http://dl.acm.org/citation.cfm?id=1283383.1283438.
  6. Davide Bilò. Almost optimal algorithms for diameter-optimally augmenting trees. Theoretical Computer Science, 931:31-48, 2022. URL: https://doi.org/10.1016/j.tcs.2022.07.028.
  7. Davide Bilò, Luciano Gualà, Stefano Leucci, and Luca Pepè Sciarria. Finding diameter-reducing shortcuts in trees. In Proceedings of the 18th Algorithms and Data Structures Symposium, WADS 2021. Google Scholar
  8. Davide Bilò, Luciano Gualà, and Guido Proietti. Improved approximability and non-approximability results for graph diameter decreasing problems. Theor. Comput. Sci., 417:12-22, 2012. Google Scholar
  9. Béla Bollobás. Extremal graph theory. Courier Corporation, 2004. Google Scholar
  10. Aléx F. Brandt, Miguel F. A. de Gaiowski, Pedro J. de Rezende, and Cid C. de Souza. Computing minimum dilation spanning trees in geometric graphs. In Proceedings of the 21st Annual International Computing and Combinatorics Conference, COCOON 2015. URL: https://doi.org/10.1007/978-3-319-21398-9_24.
  11. Leizhen Cai and Derek G. Corneil. Tree spanners. SIAM Journal on Discrete Mathematics, 8(3):359-387, 1995. URL: https://doi.org/10.1137/S0895480192237403.
  12. Otfried Cheong, Herman J. Haverkort, and Mira Lee. Computing a minimum-dilation spanning tree is NP-hard. Computational Geometry, 41(3):188-205, 2008. URL: https://doi.org/10.1016/j.comgeo.2007.12.001.
  13. Vasek Chvátal. A greedy heuristic for the set-covering problem. Mathematics of Operations Research, 4(3):233-235, 1979. URL: https://doi.org/10.1287/moor.4.3.233.
  14. Gautam Das, Paul J. Heffernan, and Giri Narasimhan. Optimally sparse spanners in 3-dimensional euclidean space. In Proceedings of the 9th Annual Symposium on Computational Geometry, SoCG 1993. URL: https://doi.org/10.1145/160985.160998.
  15. Erik D. Demaine and Morteza Zadimoghaddam. Minimizing the diameter of a network using shortcut edges. In Proceedings of the 12th Scandinavian Workshop on Algorithm Theory, SWAT 2010. URL: https://doi.org/10.1007/978-3-642-13731-0_39.
  16. Yuval Emek and David Peleg. Approximating minimum max-stretch spanning trees on unweighted graphs. SIAM Journal of Computing, 38(5):1761-1781, 2008. URL: https://doi.org/10.1137/060666202.
  17. David Eppstein. Spanning trees and spanners. In Handbook of Computational Geometry, pages 425-461. Elsevier, 2000. URL: https://doi.org/10.1016/b978-044482537-7/50010-3.
  18. David Eppstein and Kevin A. Wortman. Minimum dilation stars. In Proceedings of the 21st Annual Symposium on Computational Geometry, SoCG 2005. URL: https://doi.org/10.1145/1064092.1064142.
  19. Paul Erdős. On some extremal problems in graph theory. Israel Journal of Mathematics, 3:113-116, 1965. Google Scholar
  20. Mohammad Farshi, Panos Giannopoulos, and Joachim Gudmundsson. Improving the stretch factor of a geometric network by edge augmentation. SIAM Journal of Computing, 38(1):226-240, 2008. URL: https://doi.org/10.1137/050635675.
  21. Sándor P. Fekete and Jana Kremer. Tree spanners in planar graphs. Discrete Applied Mathematics, 108(1-2):85-103, 2001. URL: https://doi.org/10.1016/S0166-218X(00)00226-2.
  22. Arnold Filtser and Shay Solomon. The greedy spanner is existentially optimal. SIAM Journal on Computing, 49(2):429-447, 2020. URL: https://doi.org/10.1137/18M1210678.
  23. Fedor V. Fomin, Petr A. Golovach, and Erik Jan van Leeuwen. Spanners of bounded degree graphs. Information Processing Letters, 111(3):142-144, 2011. URL: https://doi.org/10.1016/j.ipl.2010.10.021.
  24. Fabrizio Frati, Serge Gaspers, Joachim Gudmundsson, and Luke Mathieson. Augmenting graphs to minimize the diameter. Algorithmica, 72(4):995-1010, 2015. URL: https://doi.org/10.1007/s00453-014-9886-4.
  25. Panos Giannopoulos, Christian Knauer, and Dániel Marx. Minimum-dilation tour (and path) is NP-hard. In Proceedings of the 23rd European Workshop on Computational Geometry, EuroCG 2007. Google Scholar
  26. Ulrike Große, Christian Knauer, Fabian Stehn, Joachim Gudmundsson, and Michiel H. M. Smid. Fast algorithms for diameter-optimally augmenting paths and trees. International Journal of Foundations of Computer Science, 30(2):293-313, 2019. URL: https://doi.org/10.1142/S0129054119500060.
  27. Joachim Gudmundsson and Yuan Sha. Algorithms for radius-optimally augmenting trees in a metric space. In Proceedings of the 17th Algorithms and Data Structures Symposium, WADS 2021. URL: https://doi.org/10.1007/978-3-030-83508-8_33.
  28. Joachim Gudmundsson, Yuan Sha, and Fan Yao. Augmenting graphs to minimize the radius. In Proceedings of the 32nd International Symposium on Algorithms and Computation, ISAAC 2021. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2021.45.
  29. Joachim Gudmundsson and Sampson Wong. Improving the dilation of a metric graph by adding edges. ACM Transactions on Algorithms, 18(3):20:1-20:20, 2022. URL: https://doi.org/10.1145/3517807.
  30. Christopher Johnson and Haitao Wang. A linear-time algorithm for radius-optimally augmenting paths in a metric space. Computational Geometry, 96:101759, 2021. URL: https://doi.org/10.1016/j.comgeo.2021.101759.
  31. Karthik C. S., Bundit Laekhanukit, and Pasin Manurangsi. On the parameterized complexity of approximating dominating set. J. ACM, 66(5):33:1-33:38, 2019. URL: https://doi.org/10.1145/3325116.
  32. Hung Le and Shay Solomon. Truly optimal Euclidean spanners. In Proceedings of the 60th Annual Symposium on Foundations of Computer Science, FOCS 2019. URL: https://doi.org/10.1109/FOCS.2019.00069.
  33. Stefano Leucci. Graph spanners. Lecture Notes for the Course on “Distributed and Sequential Graph Algorithms”, 2019. Google Scholar
  34. Lan Lin and Yixun Lin. Optimality computation of the minimum stretch spanning tree problem. Applied Mathematics and Computation, 386:125502, 2020. URL: https://doi.org/10.1016/j.amc.2020.125502.
  35. Jun Luo and Christian Wulff-Nilsen. Computing best and worst shortcuts of graphs embedded in metric spaces. In Proceedings of the 19th International Symposium on Algorithms and Computation, ISAAC 2008. URL: https://doi.org/10.1007/978-3-540-92182-0_67.
  36. Wolfgang Mulzer. Minimum dilation triangulations for the regular n-gon. Master’s thesis Freie Universität Berlin, Germany, 2004. Google Scholar
  37. Giri Narasimhan and Michiel H. M. Smid. Geometric spanner networks. Cambridge University Press, 2007. Google Scholar
  38. David Peleg. Low stretch spanning trees. In Proceedings of the 27th International Symposium of Mathematical Foundations of Computer Science, MFCS 2002. URL: https://doi.org/10.1007/3-540-45687-2_5.
  39. Mikkel Thorup and Uri Zwick. Approximate distance oracles. Journal of the ACM, 52(1):1-24, 2005. URL: https://doi.org/10.1145/1044731.1044732.
  40. Haitao Wang. An improved algorithm for diameter-optimally augmenting paths in a metric space. Computational Geometry, 75:11-21, 2018. URL: https://doi.org/10.1016/j.comgeo.2018.06.004.
  41. Haitao Wang and Yiming Zhao. A linear-time algorithm for discrete radius optimally augmenting paths in a metric space. International Journal of Computational Geometry and Applications, 30(3&4):167-182, 2020. URL: https://doi.org/10.1142/S0218195920500089.
  42. Haitao Wang and Yiming Zhao. Algorithms for diameters of unicycle graphs and diameter-optimally augmenting trees. Theoretical Computer Science, 890:192-209, 2021. URL: https://doi.org/10.1016/j.tcs.2021.09.014.
  43. Christian Wulff-Nilsen. Computing the dilation of edge-augmented graphs in metric spaces. Computational Geometry, 43(2):68-72, 2010. URL: https://doi.org/10.1016/j.comgeo.2009.03.008.
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