Document Open Access Logo

Directed Buy-At-Bulk Spanners

Authors Elena Grigorescu , Nithish Kumar , Young-San Lin

PDF HTML 5

Files

Thumbnail PDF
PDF
LIPIcs.APPROX-RANDOM.2025.22.pdf
  • Filesize: 1.08 MB
  • 24 pages
HTML5 Logo HTML (experimental)
LIPIcs.APPROX-RANDOM.2025.22.html

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Online algorithms
  • Theory of computation → Routing and network design problems
  • Theory of computation → Rounding techniques
Keywords
  • buy-at-bulk spanners
  • minimum density junction tree
  • resource constrained shortest path

Metrics

Abstract

We present a framework that unifies directed buy-at-bulk network design and directed spanner problems, namely, buy-at-bulk spanners. The goal is to find a minimum-cost routing solution for network design problems that captures economies at scale, while satisfying demands and distance constraints for terminal pairs. A more restricted version of this problem was shown to be O(2^{log^{1-ε} n})-hard to approximate, where n is the number of vertices, under a standard complexity assumption, by Elkin and Peleg (Theory of Computing Systems, 2007). Our results for buy-at-bulk spanners are the following.
- When the edge lengths are integral with magnitude polynomial in n we present:  
1) An Õ(n^{4/5 + ε})-approximation polynomial-time randomized algorithm for uniform demands. 
2) An Õ(k^{1/2 + ε})-approximation polynomial-time randomized algorithm for general demands, where k is the number of terminal pairs. This can be improved to an Õ(k^{ε})-approximation algorithm for the single-source problem. The same approximation ratios hold in the online setting. 
- When the edge lengths are rational and well-conditioned, we present an Õ(k^{1/2 + ε})-approximation polynomial-time randomized algorithm that may slightly violate the distance constraints. The result can be improved to an Õ(k^ε)-approximation algorithm for the single-source problem. The same approximation ratios hold for the online setting when the condition number is given in advance. 
To the best of our knowledge, these are the first sublinear factor approximation algorithms for directed buy-at-bulk spanners. We allow the edge lengths to be negative and the demands to be non-unit, unlike the previous literature. Our approximation ratios match the state-of-the-art ratios in special cases, namely, buy-at-bulk network design by Antonakopoulos (WAOA, 2010) and (online) weighted spanners by Grigorescu, Kumar, and Lin (APPROX 2023). Furthermore, we improve the competitive ratio for online buy-at-bulk by Chakrabarty, Ene, Krishnaswamy, and Panigrahi (SICOMP, 2018) by a factor of log R, where R is the ratio between the maximum demand and the minimum demand.

Cite As Get BibTex

Elena Grigorescu, Nithish Kumar, and Young-San Lin. Directed Buy-At-Bulk Spanners. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 22:1-22:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.22

Author Details

Elena Grigorescu
  • Department of Computer Science, University of Waterloo, Canada
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: Partially supported by NSF CCF-1910411, NSF CCF-2228814, NSF Award CCF-2127806 and ONR Award N00014-24-1-2695.

References

  1. Amir Abboud and Greg Bodwin. Reachability preservers: New extremal bounds and approximation algorithms. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1865-1883. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.122.
  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. URL: https://doi.org/10.1016/J.COSREV.2020.100253.
  3. Noga Alon, Baruch Awerbuch, Yossi Azar, Niv Buchbinder, and Joseph Naor. The online set cover problem. SIAM J. Comput., 39(2):361-370, 2009. URL: https://doi.org/10.1137/060661946.
  4. Matthew Andrews. Hardness of buy-at-bulk network design. In 45th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 115-124. IEEE, 2004. URL: https://doi.org/10.1109/FOCS.2004.32.
  5. Spyridon Antonakopoulos. Buy-at-bulk-related problems in network design. PhD thesis, Columbia University, 2009. Google Scholar
  6. Baruch Awerbuch. Communication-time trade-offs in network synchronization. In Proceedings of the Fourth Annual ACM Symposium on Principles of Distributed Computing, PODC ’85, pages 272-276, New York, NY, USA, 1985. Association for Computing Machinery. URL: https://doi.org/10.1145/323596.323621.
  7. Baruch Awerbuch and Yossi Azar. Buy-at-bulk network design. In Proceedings 38th Annual Symposium on Foundations of Computer Science (FOCS), pages 542-547. IEEE, 1997. URL: https://doi.org/10.1109/SFCS.1997.646143.
  8. Surender Baswana and Telikepalli Kavitha. Faster algorithms for all-pairs approximate shortest paths in undirected graphs. SIAM J. Comput., 39(7):2865-2896, 2010. URL: https://doi.org/10.1137/080737174.
  9. 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. URL: https://doi.org/10.1016/J.IC.2012.10.007.
  10. Arnab Bhattacharyya, Elena Grigorescu, Kyomin Jung, Sofya Raskhodnikova, and David P Woodruff. Transitive-closure spanners. SIAM Journal on Computing, 41(6):1380-1425, 2012. URL: https://doi.org/10.1137/110826655.
  11. Greg Bodwin and Virginia Vassilevska Williams. Better distance preservers and additive spanners. In Robert Krauthgamer, editor, SODA, pages 855-872. SIAM, 2016. URL: https://doi.org/10.1137/1.9781611974331.CH61.
  12. Deeparnab Chakrabarty, Alina Ene, Ravishankar Krishnaswamy, and Debmalya Panigrahi. Online buy-at-bulk network design. SIAM J. Comput., 47(4):1505-1528, 2018. URL: https://doi.org/10.1137/16M1117317.
  13. 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. URL: https://doi.org/10.1006/JAGM.1999.1042.
  14. Moses Charikar and Adriana Karagiozova. On non-uniform multicommodity buy-at-bulk network design. In Proceedings of the 37th Annual ACM Symposium on Theory of computing (STOC), pages 176-182, 2005. URL: https://doi.org/10.1145/1060590.1060617.
  15. C Chekuri, MT Hajiaghayi, G Kortsarz, and MR Salavatipour. Approximation algorithms for node-weighted buy-at-bulk network design. In Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1265-1274, 2007. Google Scholar
  16. 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. URL: https://doi.org/10.1145/1921659.1921664.
  17. Chandra Chekuri, Mohammad Taghi Hajiaghayi, Guy Kortsarz, and Mohammad R Salavatipour. Approximation algorithms for nonuniform buy-at-bulk network design. SIAM Journal on Computing, 39(5):1772-1798, 2010. URL: https://doi.org/10.1137/090750317.
  18. 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. URL: https://doi.org/10.1145/3381451.
  19. Julia Chuzhoy, Anupam Gupta, Joseph Naor, and Amitabh Sinha. On the approximability of some network design problems. ACM Transactions on Algorithms (TALG), 4(2):1-17, 2008. URL: https://doi.org/10.1145/1361192.1361200.
  20. Lenore Cowen and Christopher G. Wagner. Compact roundtrip routing in directed networks. J. Algorithms, 50(1):79-95, 2004. URL: https://doi.org/10.1016/J.JALGOR.2003.08.001.
  21. Michael Dinitz and Robert Krauthgamer. Directed spanners via flow-based linear programs. In STOC, pages 323-332, 2011. URL: https://doi.org/10.1145/1993636.1993680.
  22. Michael Dinitz and Zeyu Zhang. Approximating low-stretch spanners. In Proceedings of the twenty-seventh annual ACM-SIAM symposium on Discrete algorithms, pages 821-840. SIAM, 2016. URL: https://doi.org/10.1137/1.9781611974331.CH59.
  23. Yevgeniy Dodis and Sanjeev Khanna. Design networks with bounded pairwise distance. In Proceedings of the thirty-first annual ACM symposium on Theory of computing, pages 750-759, 1999. URL: https://doi.org/10.1145/301250.301447.
  24. Dorit Dor, Shay Halperin, and Uri Zwick. All-pairs almost shortest paths. SIAM J. Comput., 29(5):1740-1759, 2000. URL: https://doi.org/10.1137/S0097539797327908.
  25. Michael Elkin. Computing almost shortest paths. ACM Trans. Algorithms, 1(2):283-323, 2005. URL: https://doi.org/10.1145/1103963.1103968.
  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 Comput. Syst., 41(4):691-729, 2007. URL: https://doi.org/10.1007/S00224-006-1266-2.
  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. URL: https://doi.org/10.1016/J.JCSS.2011.05.009.
  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, 2022. 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 thirty-eighth annual ACM symposium on Theory of computing, pages 663-670, 2006. URL: https://doi.org/10.1145/1132516.1132609.
  31. Rohan Ghuge and Viswanath Nagarajan. Quasi-polynomial algorithms for submodular tree orienteering and directed network design problems. Mathematics of Operations Research, 47(2):1612-1630, 2022. URL: https://doi.org/10.1287/MOOR.2021.1181.
  32. Elena Grigorescu, Nithish Kumar, and Young-San Lin. Approximation algorithms for directed weighted spanners. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, 2023. Google Scholar
  33. Elena Grigorescu, Nithish Kumar, and Young-San Lin. Directed buy-at-bulk spanners. arXiv preprint arXiv:2404.05172, 2024. URL: https://doi.org/10.48550/arXiv.2404.05172.
  34. 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), pages 5:1-5:25. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2021.5.
  35. Sudipto Guha, Adam Meyerson, and Kamesh Munagala. A constant factor approximation for the single sink edge installation problem. SIAM Journal on Computing, 38(6):2426-2442, 2009. URL: https://doi.org/10.1137/050643635.
  36. 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, 2003. Proceedings., pages 606-615. IEEE, 2003. URL: https://doi.org/10.1109/SFCS.2003.1238233.
  37. Anupam Gupta, Amit Kumar, and Tim Roughgarden. Simpler and better approximation algorithms for network design. In Proceedings of the 35th Annual ACM Symposium on Theory of computing (STOC), pages 365-372, 2003. URL: https://doi.org/10.1145/780542.780597.
  38. Anupam Gupta, R Ravi, Kunal Talwar, and Seeun William Umboh. Last but not least: Online spanners for buy-at-bulk. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 589-599. SIAM, 2017. URL: https://doi.org/10.1137/1.9781611974782.38.
  39. 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. URL: https://doi.org/10.1145/3406325.3451053.
  40. Mohammad Taghi Hajiaghayi, Guy Kortsarz, and Mohammad R Salavatipour. Approximating buy-at-bulk and shallow-light k-steiner trees. Algorithmica, 53:89-103, 2009. URL: https://doi.org/10.1007/S00453-007-9013-X.
  41. Mohammad Taghi Hajiaghayi, Vahid Liaghat, and Debmalya Panigrahi. Online node-weighted steiner forest and extensions via disk paintings. In 2013 IEEE 54th Annual Symposium on Foundations of Computer Science (FOCS), pages 558-567. IEEE, 2013. URL: https://doi.org/10.1109/FOCS.2013.66.
  42. MohammadTaghi Hajiaghayi, Vahid Liaghat, and Debmalya Panigrahi. Near-optimal online algorithms for prize-collecting steiner problems. In Automata, Languages, and Programming: 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part I 41, pages 576-587. Springer, 2014. URL: https://doi.org/10.1007/978-3-662-43948-7_48.
  43. Refael Hassin. Approximation schemes for the restricted shortest path problem. Mathematics of Operations research, 17(1):36-42, 1992. URL: https://doi.org/10.1287/MOOR.17.1.36.
  44. Markó Horváth and Tamás Kis. Multi-criteria approximation schemes for the resource constrained shortest path problem. Optimization Letters, 12(3):475-483, 2018. URL: https://doi.org/10.1007/S11590-017-1212-Z.
  45. P Klein and R Ravi. A nearly best-possible approximation algorithm for node-weighted steiner trees. Journal of Algorithms, 1(19):104-115, 1995. Google Scholar
  46. Guy Kortsarz. On the hardness of approximating spanners. Algorithmica, 30:432-450, 2001. URL: https://doi.org/10.1007/S00453-001-0021-Y.
  47. 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. URL: https://doi.org/10.1016/S0167-6377(01)00069-4.
  48. 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. URL: https://doi.org/10.1006/JAGM.1998.0930.
  49. Adam Meyerson, Kamesh Munagala, and Serge Plotkin. Cost-distance: Two metric network design. SIAM Journal on Computing, 38(4):1648-1659, 2008. URL: https://doi.org/10.1137/050629665.
  50. Joseph Naor, Debmalya Panigrahi, and Mohit Singh. Online node-weighted steiner tree and related problems. In 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science (FOCS), pages 210-219. IEEE, 2011. URL: https://doi.org/10.1109/FOCS.2011.65.
  51. 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, SODA, pages 1374-1392. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.91.
  52. 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.
  53. David Peleg and Jeffrey D. Ullman. An optimal synchronizer for the hypercube. SIAM J. Comput., 18(4):740-747, 1989. URL: https://doi.org/10.1137/0218050.
  54. Liam Roditty, Mikkel Thorup, and Uri Zwick. Roundtrip spanners and roundtrip routing in directed graphs. ACM Trans. Algorithms, 4(3):29:1-29:17, 2008. URL: https://doi.org/10.1145/1367064.1367069.
  55. F Sibel Salman, Joseph Cheriyan, Ramamoorthi Ravi, and Sairam Subramanian. Approximating the single-sink link-installation problem in network design. SIAM Journal on Optimization, 11(3):595-610, 2001. URL: https://doi.org/10.1137/S1052623497321432.
  56. Xiangkun Shen and Viswanath Nagarajan. Online covering with l_q-norm objectives and applications to network design. Mathematical Programming, 184, 2020. URL: https://doi.org/10.1007/S10107-019-01409-9.
  57. Kunal Talwar. The single-sink buy-at-bulk lp has constant integrality gap. In International Conference on Integer Programming and Combinatorial Optimization (IPCO), pages 475-486. Springer, 2002. URL: https://doi.org/10.1007/3-540-47867-1_33.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact