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Approximation Algorithms for Directed Weighted Spanners

Authors Elena Grigorescu , Nithish Kumar, Young-San Lin

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Author Details

Elena Grigorescu
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA
Nithish Kumar
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA
Young-San Lin
  • Melbourne Business School, Australia

Funding

  • Grigorescu, Elena: Supported in part by NSF CCF-1910659, NSF CCF-1910411, and NSF CCF-2228814.
  • Kumar, Nithish: Supported in part by NSF CCF-1910411 and NSF CCF-2228814.

Acknowledgements

We thank several anonymous reviewers for their valuable comments and suggestions that improved the quality of the writeup.

Cite AsGet BibTex

Elena Grigorescu, Nithish Kumar, and Young-San Lin. Approximation Algorithms for Directed Weighted Spanners. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 8:1-8:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.8

Abstract

In the pairwise weighted spanner problem, the input consists of a weighted directed graph on n vertices, where each edge is assigned both a cost and a length. Furthermore, we are given k terminal vertex pairs and a distance constraint for each pair. The goal is to find a minimum-cost subgraph in which the distance constraints are satisfied. We study the weighted spanner problem, in which the edges have positive integral lengths of magnitudes that are polynomial in n, while the costs are arbitrary non-negative rational numbers. Our results include the following in the classical offline setting: - An Õ(n^{4/5 + ε})-approximation algorithm for the weighted pairwise spanner problem. When the edges have unit costs and lengths, the best previous algorithm gives an Õ(n^{3/5 + ε})-approximation, due to Chlamtáč, Dinitz, Kortsarz, and Laekhanukit (Transactions on Algorithms, 2020). - An Õ(n^{1/2+ε})-approximation algorithm for the weighted spanner problem when the terminal pairs consist of all vertex pairs and the distances must be preserved exactly. When the edges have unit costs and arbitrary positive lengths, the best previous algorithm gives an Õ(n^{1/2})-approximation for the all-pair spanner problem, due to Berman, Bhattacharyya, Makarychev, Raskhodnikova, and Yaroslavtsev (Information and Computation, 2013). We also prove the first results for the weighted spanners in the online setting. Our results include the following: - An Õ(k^{1/2 + ε})-competitive algorithm for the online weighted pairwise spanner problem. The state-of-the-art results are an Õ(n^{4/5})-competitive algorithm when edges have unit costs and arbitrary positive lengths, and a min{Õ(k^{1/2 + ε}), Õ(n^{2/3 + ε})}-competitive algorithm when edges have unit costs and lengths, due to Grigorescu, Lin, and Quanrud (APPROX, 2021). - An Õ(k^ε)-competitive algorithm for the online weighted single-source (or single-sink) spanner problem. Without distance constraints, this problem is equivalent to the online directed Steiner tree problem. The best previous algorithm for online directed Steiner trees is an Õ(k^ε)-competitive algorithm, due to Chakrabarty, Ene, Krishnaswamy, and Panigrahi (SICOMP, 2018). Our online results also imply efficient approximation algorithms for the corresponding offline problems. To the best of our knowledge, these are the first approximation (online) polynomial-time algorithms with sublinear approximation (competitive) ratios for the weighted spanner problems.

Subject Classification

ACM Subject Classification
  • Theory of computation → Online algorithms
  • Theory of computation → Routing and network design problems
  • Theory of computation → Rounding techniques
Keywords
  • directed weighted spanners
  • linear programming
  • junction tree

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References

  1. Amir Abboud and Greg Bodwin. Reachability preservers: New extremal bounds and approximation algorithms. In Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1865-1883. SIAM, 2018. Google Scholar
  2. Reyan Ahmed, Greg Bodwin, Faryad Darabi Sahneh, Keaton Hamm, Mohammad Javad Latifi Jebelli, Stephen Kobourov, and Richard Spence. Graph spanners: A tutorial review. Computer Science Review, 37:100253, 2020. Google Scholar
  3. Noga Alon, Baruch Awerbuch, Yossi Azar, Niv Buchbinder, and Joseph Naor. A general approach to online network optimization problems. ACM Transactions on Algorithms (TALG), 2(4):640-660, 2006. Google Scholar
  4. Noga Alon and Baruch Schieber. Optimal preprocessing for answering on-line product queries. Technical report, Tel-Aviv University, 1987. Google Scholar
  5. Spyridon Antonakopoulos. Approximating directed buy-at-bulk network design. In International Workshop on Approximation and Online Algorithms, pages 13-24. Springer, 2010. Google Scholar
  6. Pranjal Awasthi, Madhav Jha, Marco Molinaro, and Sofya Raskhodnikova. Testing lipschitz functions on hypergrid domains. Algorithmica, 74(3):1055-1081, 2016. Google Scholar
  7. Baruch Awerbuch. Communication-time trade-offs in network synchronization. In Proceedings of the 4th Annual ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing (PODC), pages 272-276, New York, NY, USA, 1985. Google Scholar
  8. Surender Baswana and Telikepalli Kavitha. Faster algorithms for all-pairs approximate shortest paths in undirected graphs. SIAM Journal on Computing, 39(7):2865-2896, 2010. Google Scholar
  9. MohammadHossein Bateni and MohammadTaghi Hajiaghayi. Euclidean prize-collecting steiner forest. Algorithmica, 62(3-4):906-929, 2012. Google Scholar
  10. Piotr Berman, Arnab Bhattacharyya, Konstantin Makarychev, Sofya Raskhodnikova, and Grigory Yaroslavtsev. Approximation algorithms for spanner problems and directed steiner forest. Information and Computation, 222:93-107, 2013. Google Scholar
  11. Arnab Bhattacharyya, Elena Grigorescu, Kyomin Jung, Sofya Raskhodnikova, and David P Woodruff. Transitive-closure spanners. SIAM Journal on Computing, 41(6):1380-1425, 2012. Google Scholar
  12. Greg Bodwin. New results on linear size distance preservers. SIAM Journal on Computing, 50(2):662-673, 2021. Google Scholar
  13. Glencora Borradaile, Philip N Klein, and Claire Mathieu. A polynomial-time approximation scheme for euclidean steiner forest. ACM Transactions on Algorithms (TALG), 11(3):1-20, 2015. Google Scholar
  14. Deeparnab Chakrabarty, Alina Ene, Ravishankar Krishnaswamy, and Debmalya Panigrahi. Online buy-at-bulk network design. SIAM Journal on Computing, 47(4):1505-1528, 2018. Google Scholar
  15. Moses Charikar, Chandra Chekuri, To-Yat Cheung, Zuo Dai, Ashish Goel, Sudipto Guha, and Ming Li. Approximation algorithms for directed steiner problems. Journal of Algorithms, 33(1):73-91, 1999. Google Scholar
  16. Shuchi Chawla, Tim Roughgarden, and Mukund Sundararajan. Optimal cost-sharing mechanisms for steiner forest problems. In International Workshop on Internet and Network Economics, pages 112-123. Springer, 2006. Google Scholar
  17. Shiri Chechik. Approximate distance oracles with improved bounds. In Rocco A. Servedio and Ronitt Rubinfeld, editors, Proceedings of the 47th Annual ACM Symposium on Theory of Computing (STOC), pages 1-10. ACM, 2015. Google Scholar
  18. Chandra Chekuri, Guy Even, Anupam Gupta, and Danny Segev. Set connectivity problems in undirected graphs and the directed steiner network problem. ACM Transactions on Algorithms (TALG), 7(2):1-17, 2011. Google Scholar
  19. Eden Chlamtáč, Michael Dinitz, Guy Kortsarz, and Bundit Laekhanukit. Approximating spanners and directed steiner forest: Upper and lower bounds. ACM Transactions on Algorithms (TALG), 16(3):1-31, 2020. Google Scholar
  20. Lenore Cowen and Christopher G. Wagner. Compact roundtrip routing in directed networks. Journal on Algorithms, 50(1):79-95, 2004. Google Scholar
  21. Michael Dinitz and Robert Krauthgamer. Directed spanners via flow-based linear programs. In Proceedings of the 43rd Annual ACM Symposium on Theory of Computing (STOC), pages 323-332, 2011. Google Scholar
  22. Michael Dinitz and Robert Krauthgamer. Fault-tolerant spanners: better and simpler. In Proceedings of the 30th Annual ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing (PODC), pages 169-178, 2011. Google Scholar
  23. Michael Dinitz and Zeyu Zhang. Approximating low-stretch spanners. In Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete algorithms (SODA), pages 821-840. SIAM, 2016. Google Scholar
  24. Dorit Dor, Shay Halperin, and Uri Zwick. All-pairs almost shortest paths. SIAM Journal on Computing, 29(5):1740-1759, 2000. Google Scholar
  25. Michael Elkin. Computing almost shortest paths. ACM Transactions on Algorithms (TALG), 1(2):283-323, 2005. Google Scholar
  26. Michael Elkin and David Peleg. The client-server 2-spanner problem with applications to network design. In Francesc Comellas, Josep Fàbrega, and Pierre Fraigniaud, editors, SIROCCO 8, Proceedings of the 8th International Colloquium on Structural Information and Communication Complexity, Vall de Núria, Girona-Barcelona, Catalonia, Spain, 27-29 June, 2001, volume 8 of Proceedings in Informatics, pages 117-132. Carleton Scientific, 2001. Google Scholar
  27. Michael Elkin and David Peleg. The hardness of approximating spanner problems. Theory of Computing Systems, 41(4):691-729, 2007. Google Scholar
  28. Moran Feldman, Guy Kortsarz, and Zeev Nutov. Improved approximation algorithms for directed steiner forest. Journal of Computer and System Sciences, 78(1):279-292, 2012. Google Scholar
  29. Arnold Filtser. Hop-constrained metric embeddings and their applications. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 492-503. IEEE, 2021. Google Scholar
  30. Lisa Fleischer, Jochen Könemann, Stefano Leonardi, and Guido Schäfer. Simple cost sharing schemes for multicommodity rent-or-buy and stochastic steiner tree. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC), pages 663-670, 2006. Google Scholar
  31. Fedor V Fomin, Petr A Golovach, William Lochet, Pranabendu Misra, Saket Saurabh, and Roohani Sharma. Parameterized complexity of directed spanner problems. Algorithmica, 84(8):2292-2308, 2022. Google Scholar
  32. Elena Grigorescu, Nithish Kumar, and Young-San Lin. Approximation algorithms for directed weighted spanners, 2023. URL: https://arxiv.org/abs/2307.02774.
  33. Elena Grigorescu, Young-San Lin, and Kent Quanrud. Online directed spanners and steiner forests. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  34. Anupam Gupta, Amit Kumar, Martin Pál, and Tim Roughgarden. Approximation via cost-sharing: a simple approximation algorithm for the multicommodity rent-or-buy problem. In 44th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2003. Proceedings., pages 606-615. IEEE, 2003. Google Scholar
  35. Bernhard Haeupler, D Ellis Hershkowitz, and Goran Zuzic. Tree embeddings for hop-constrained network design. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 356-369, 2021. Google Scholar
  36. Mohammad Taghi Hajiaghayi, Guy Kortsarz, and Mohammad R Salavatipour. Approximating buy-at-bulk and shallow-light k-steiner trees. Algorithmica, 53:89-103, 2009. Google Scholar
  37. Refael Hassin. Approximation schemes for the restricted shortest path problem. Mathematics of Operations research, 17(1):36-42, 1992. Google Scholar
  38. Markó Horváth and Tamás Kis. Multi-criteria approximation schemes for the resource constrained shortest path problem. Optimization Letters, 12(3):475-483, 2018. Google Scholar
  39. Samir Khuller, Balaji Raghavachari, and Neal Young. Approximating the minimum equivalent digraph. SIAM Journal on Computing, 24(4):859-872, 1995. Google Scholar
  40. Vikram Khurana, Jian Peng, Chee Yeun Chung, Pavan K Auluck, Saranna Fanning, Daniel F Tardiff, Theresa Bartels, Martina Koeva, Stephen W Eichhorn, Hadar Benyamini, et al. Genome-scale networks link neurodegenerative disease genes to α-synuclein through specific molecular pathways. Cell systems, 4(2):157-170, 2017. Google Scholar
  41. Shimon Kogan and Merav Parter. Having hope in hops: New spanners, preservers and lower bounds for hopsets. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 766-777. IEEE, 2022. Google Scholar
  42. Jochen Könemann, Stefano Leonardi, Guido Schäfer, and Stefan van Zwam. From primal-dual to cost shares and back: a stronger lp relaxation for the steiner forest problem. In International Colloquium on Automata, Languages, and Programming, pages 930-942. Springer, 2005. Google Scholar
  43. Jochen Könemann, Stefano Leonardi, Guido Schäfer, and Stefan HM van Zwam. A group-strategyproof cost sharing mechanism for the steiner forest game. SIAM Journal on Computing, 37(5):1319-1341, 2008. Google Scholar
  44. Guy Kortsarz. On the hardness of approximating spanners. Algorithmica, 30:432-450, 2001. Google Scholar
  45. Dean H Lorenz and Danny Raz. A simple efficient approximation scheme for the restricted shortest path problem. Operations Research Letters, 28(5):213-219, 2001. Google Scholar
  46. Madhav V Marathe, Ravi Sundaram, SS Ravi, Daniel J Rosenkrantz, and Harry B Hunt III. Bicriteria network design problems. Journal of algorithms, 28(1):142-171, 1998. Google Scholar
  47. Jakub Pachocki, Liam Roditty, Aaron Sidford, Roei Tov, and Virginia Vassilevska Williams. Approximating cycles in directed graphs: Fast algorithms for girth and roundtrip spanners. In Artur Czumaj, editor, Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete algorithms (SODA), pages 1374-1392. SIAM, 2018. Google Scholar
  48. Mihai Patrascu and Liam Roditty. Distance oracles beyond the Thorup-Zwick bound. SIAM Journal on Computing, 43(1):300-311, 2014. Google Scholar
  49. David Peleg and Alejandro A. Schäffer. Graph spanners. Journal of Graph Theory, 13(1):99-116, 1989. URL: https://doi.org/10.1002/jgt.3190130114.
  50. David Peleg and Jeffrey D. Ullman. An optimal synchronizer for the hypercube. SIAM Journal on Computing, 18(4):740-747, 1989. Google Scholar
  51. Leila Pirhaji, Pamela Milani, Mathias Leidl, Timothy Curran, Julian Avila-Pacheco, Clary B Clish, Forest M White, Alan Saghatelian, and Ernest Fraenkel. Revealing disease-associated pathways by network integration of untargeted metabolomics. Nature methods, 13(9):770-776, 2016. Google Scholar
  52. Liam Roditty, Mikkel Thorup, and Uri Zwick. Roundtrip spanners and roundtrip routing in directed graphs. ACM Transactions on Algorithms (TALG), 4(3):29:1-29:17, 2008. Google Scholar
  53. Tim Roughgarden and Mukund Sundararajan. Optimal efficiency guarantees for network design mechanisms. In International Conference on Integer Programming and Combinatorial Optimization, pages 469-483. Springer, 2007. Google Scholar
  54. Adrian Vetta. Approximating the minimum strongly connected subgraph via a matching lower bound. In Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete algorithms (SODA), pages 417-426, 2001. Google Scholar
  55. Andrew Chi-Chih Yao. Space-time tradeoff for answering range queries (extended abstract). In Proceedings of the 14th Annual ACM Symposium on Theory of Computing (STOC), 1982. Google Scholar
  56. Alexander Zelikovsky. A series of approximation algorithms for the acyclic directed steiner tree problem. Algorithmica, 18(1):99-110, 1997. Google Scholar
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