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Approximation Algorithms for the Airport and Railway Problem

Authors Mohammad R. Salavatipour , Lijiangnan Tian

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Author Details

Mohammad R. Salavatipour
  • Department of Computer Science, University of Alberta, Edmonton, Canada
Lijiangnan Tian
  • Department of Computer Science, University of Alberta, Edmonton, Canada

Funding

  • Salavatipour, Mohammad R.: Supported by NSERC DG.
  • Tian, Lijiangnan: Supported by first authors NSERC.

Acknowledgements

We want to thank Zachary Friggstad for his comments that improved and simplified Theorem 1. We also talked to Mohsen Rezapour for some initial discussions.

Cite AsGet BibTex

Mohammad R. Salavatipour and Lijiangnan Tian. Approximation Algorithms for the Airport and Railway Problem. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 40:1-40:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SWAT.2024.40

Abstract

In this paper, we present approximation algorithms for the airport and railway problem (AR) on several classes of graphs. The AR problem, introduced by [Anna Adamaszek et al., 2016], is a combination of the Capacitated Facility Location problem (CFL) and the network design problem. An AR instance consists of a set of points (cities) V in a metric d(.,.), each of which is associated with a non-negative cost f_v and a number k, which represent respectively the cost of establishing an airport (facility) in the corresponding point, and the universal airport capacity. A feasible solution is a network of airports and railways providing services to all cities without violating any capacity, where railways are edges connecting pairs of points, with their costs equivalent to the distance between the respective points. The objective is to find such a network with the least cost. In other words, find a forest, each component having at most k points and one open facility, minimizing the total cost of edges and airport opening costs. Adamaszek et al. [Anna Adamaszek et al., 2016] presented a PTAS for AR in the two-dimensional Euclidean metric ℝ² with a uniform opening cost. In subsequent work [Anna Adamaszek et al., 2018] presented a bicriteria 4/3 (2+1/α)-approximation algorithm for AR with non-uniform opening costs but violating the airport capacity by a factor of 1+α, i.e. (1+α)k capacity where 0 < α ≤ 1, a (2+k/(k-1)+ε)-approximation algorithm and a bicriteria Quasi-Polynomial Time Approximation Scheme (QPTAS) for the same problem in the Euclidean plane ℝ². In this work, we give a 2-approximation for AR with a uniform opening cost for general metrics and an O(log n)-approximation for non-uniform opening costs. We also give a QPTAS for AR with a uniform opening cost in graphs of bounded treewidth and a QPTAS for a slightly relaxed version in the non-uniform setting. The latter implies O(1)-approximation on graphs of bounded doubling dimensions, graphs of bounded highway dimensions and planar graphs in quasi-polynomial time.

Subject Classification

ACM Subject Classification
  • Theory of computation
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Facility Location
  • Approximation Algorithms
  • Dynamic Programming

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References

  1. Anna Adamaszek, Antonios Antoniadis, Amit Kumar, and Tobias Mömke. Approximating Airports and Railways. In Rolf Niedermeier and Brigitte Vallée, editors, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018), volume 96 of Leibniz International Proceedings in Informatics (LIPIcs), pages 5:1-5:13, Dagstuhl, Germany, 2018. Schloss Dagstuhl-Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.STACS.2018.5.
  2. Anna Adamaszek, Antonios Antoniadis, and Tobias Mömke. Airports and Railways: Facility Location Meets Network Design. In Nicolas Ollinger and Heribert Vollmer, editors, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016), volume 47 of Leibniz International Proceedings in Informatics (LIPIcs), pages 6:1-6:14, Dagstuhl, Germany, 2016. Schloss Dagstuhl-Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.STACS.2016.6.
  3. Jannis Blauth, Vera Traub, and Jens Vygen. Improving the approximation ratio for capacitated vehicle routing. Mathematical Programming, 197(2):451-497, 2023. URL: https://doi.org/10.1007/S10107-022-01841-4.
  4. Hans L. Bodlaender and Torben Hagerup. Parallel Algorithms with Optimal Speedup for Bounded Treewidth. SIAM Journal on Computing, 27(6):1725-1746, 1998. URL: https://doi.org/10.1137/S0097539795289859.
  5. Vincent Cohen-Addad, Hung Le, Marcin Pilipczuk, and Michał Pilipczuk. Planar and Minor-Free Metrics Embed into Metrics of Polylogarithmic Treewidth with Expected Multiplicative Distortion Arbitrarily Close to 1, 2023. URL: https://arxiv.org/abs/2304.07268.
  6. Aparna Das and Claire Mathieu. A Quasi-polynomial Time Approximation Scheme for Euclidean Capacitated Vehicle Routing. In Moses Charikar, editor, Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, Austin, Texas, USA, January 17-19, 2010, pages 390-403. SIAM, 2010. URL: https://doi.org/10.1137/1.9781611973075.33.
  7. Jack Edmonds. Matroid Intersection. In P.L. Hammer, E.L. Johnson, and B.H. Korte, editors, Discrete Optimization I, volume 4 of Annals of Discrete Mathematics, pages 39-49. Elsevier, 1979. URL: https://doi.org/10.1016/S0167-5060(08)70817-3.
  8. Andreas Emil Feldmann, Wai Shing Fung, Jochen Könemann, and Ian Post. A (1+ε)-Embedding of Low Highway Dimension Graphs into Bounded Treewidth Graphs. SIAM Journal on Computing, 47(4):1667-1704, January 2018. URL: https://doi.org/10.1137/16m1067196.
  9. Zachary Friggstad, Ramin Mousavi, Mirmahdi Rahgoshay, and Mohammad R. Salavatipour. Improved Approximations for Capacitated Vehicle Routing with Unsplittable Client Demands. In Karen I. Aardal and Laura Sanità, editors, Integer Programming and Combinatorial Optimization - 23rd International Conference, IPCO 2022, Eindhoven, The Netherlands, June 27-29, 2022, Proceedings, volume 13265 of Lecture Notes in Computer Science, pages 251-261. Springer, 2022. URL: https://doi.org/10.1007/978-3-031-06901-7_19.
  10. Aditya Jayaprakash and Mohammad R. Salavatipour. Approximation Schemes for Capacitated Vehicle Routing on Graphs of Bounded Treewidth, Bounded Doubling, or Highway Dimension. ACM Transactions on Algorithms, 19(2), March 2023. URL: https://doi.org/10.1145/3582500.
  11. Raja Jothi and Balaji Raghavachari. Approximation Algorithms for the Capacitated Minimum Spanning Tree Problem and Its Variants in Network Design. ACM Transactions on Algorithms, 1(2):265-282, October 2005. URL: https://doi.org/10.1145/1103963.1103967.
  12. Mong-Jen Kao. Improved LP-based approximation algorithms for facility location with hard capacities. arXiv preprint, 2021. URL: https://arxiv.org/abs/2102.06613.
  13. M. Reza Khani and Mohammad R. Salavatipour. Improved Approximation Algorithms for the Min-max Tree Cover and Bounded Tree Cover Problems. Algorithmica, 69(2):443-460, 2014. URL: https://doi.org/10.1007/S00453-012-9740-5.
  14. Jens Maßberg and Jens Vygen. Approximation Algorithms for a Facility Location Problem with Service Capacities. ACM Transactions on Algorithms, 4(4), August 2008. URL: https://doi.org/10.1145/1383369.1383381.
  15. Claire Mathieu and Hang Zhou. A PTAS for Capacitated Vehicle Routing on Trees. ACM Transactions on Algorithms, 19(2):17:1-17:28, 2023. URL: https://doi.org/10.1145/3575799.
  16. Runjie Miao and Jinjiang Yuan. A note on LP-based approximation algorithms for capacitated facility location problem. Theoretical Computer Science, 932:31-40, 2022. URL: https://doi.org/10.1016/j.tcs.2022.08.002.
  17. R. Ravi and Amitabh Sinha. Approximation Algorithms for Problems Combining Facility Location and Network Design. Operations Research, 54(1):73-81, 2006. URL: https://EconPapers.repec.org/RePEc:inm:oropre:v:54:y:2006:i:1:p:73-81.
  18. Kunal Talwar. Bypassing the Embedding: Algorithms for Low Dimensional Metrics. In Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing, STOC '04, pages 281-290, New York, NY, USA, 2004. Association for Computing Machinery. URL: https://doi.org/10.1145/1007352.1007399.
  19. Lijiangnan Tian. Approximation Schemes for the Airport and Railway Problem. Master’s thesis, Department of Computing Science, Faculty of Science, University of Alberta, 2023. Google Scholar
  20. Vera Traub and Thorben Tröbst. A Fast (2 + 2/7)-Approximation Algorithm for Capacitated Cycle Covering. Mathematical Programming, 192(1):497-518, 2022. URL: https://doi.org/10.1007/S10107-021-01678-3.
  21. Wei Yu, Zhaohui Liu, and Xiaoguang Bao. New Approximation Algorithms for the Minimum Cycle Cover Problem. Theoretical Computer Science, 793:44-58, 2019. URL: https://doi.org/10.1016/J.TCS.2019.04.009.
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