Distributed Distance-Bounded Network Design Through Distributed Convex Programming

Authors Michael Dinitz, Yasamin Nazari



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Michael Dinitz
Yasamin Nazari

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Michael Dinitz and Yasamin Nazari. Distributed Distance-Bounded Network Design Through Distributed Convex Programming. In 21st International Conference on Principles of Distributed Systems (OPODIS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 95, pp. 5:1-5:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.OPODIS.2017.5

Abstract

Solving linear programs is often a challenging task in distributed settings. While there are good algorithms for solving packing and covering linear programs in a distributed manner (Kuhn et al. 2006), this is essentially the only class of linear programs for which such an algorithm is known. In this work we provide a distributed algorithm for solving a different class of convex programs which we call “distance-bounded network design convex programs”. These can be thought of as relaxations of network design problems in which the connectivity requirement includes a distance constraint (most notably, graph spanners). Our algorithm runs in O((D/ε) log n) rounds in the LOCAL model and with high probability finds a (1+ε)-approximation to the optimal LP solution for any 0 < ε ≤ 1, where D is the largest distance constraint.
While solving linear programs in a distributed setting is interesting in its own right, this class of convex programs is particularly important because solving them is often a crucial step when designing approximation algorithms. Hence we almost immediately obtain new and improved distributed approximation algorithms for a variety of network design problems, including Basic 3- and 4-Spanner, Directed k-Spanner, Lowest Degree k-Spanner, and Shallow-Light Steiner Network Design with a spanning demand graph. Our algorithms do not require any “heavy” computation and essentially match the best-known centralized approximation algorithms, while previous approaches which do not use heavy computation give approximations which are worse than the best-known centralized bounds.

Subject Classification

Keywords
  • distributed algorithms
  • approximation algorithms
  • convex programming

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References

  1. Amy Babay, Emily Wagner, Michael Dinitz, and Yair Amir. Timely, reliable, and cost-effective internet transport service using dissemination graphs. In 37th IEEE International Conference on Distributed Computing Systems, (ICDCS), pages 1-12, 2017. Google Scholar
  2. Leonid Barenboim, Michael Elkin, and Cyril Gavoille. A fast network-decomposition algorithm and its applications to constant-time distributed computation. Theoretical Computer Science, 2016. Google Scholar
  3. Yair Bartal. Probabilistic approximations of metric spaces and its algorithmic applications. In FOCS'96, pages 184-193, 1996. Google Scholar
  4. Yair Bartal, John W. Byers, and Danny Raz. Global optimization using local information with applications to flow control. In FOCS, pages 303-312, 1997. Google Scholar
  5. Piotr Berman, Arnab Bhattacharyya, Konstantin Makarychev, Sofya Raskhodnikova, and Grigory Yaroslavtsev. Improved approximation for the directed spanner problem. In ICALP Part I, pages 1-12, 2011. Google Scholar
  6. Eden Chlamtác and Michael Dinitz. Lowest-degree k-spanner: Approximation and hardness. Theory of Computing, 12(1):1-29, 2016. Google Scholar
  7. Eden Chlamtác, Michael Dinitz, Guy Kortsarz, and Bundit Laekhanukit. Approximating spanners and directed steiner forest: Upper and lower bounds. In SODA, 2017. Google Scholar
  8. Michael Dinitz, Guy Kortsarz, and Ran Raz. Label cover instances with large girth and the hardness of approximating basic k-spanner. ACM Trans. Algorithms, 12(2):1-16, 2016. Google Scholar
  9. Michael Dinitz and Robert Krauthgamer. Directed spanners via flow-based linear programs. In STOC'11, pages 323-332, 2011. Google Scholar
  10. Michael Dinitz and Robert Krauthgamer. Fault-tolerant spanners: better and simpler. In PODC'11, pages 169-178, 2011. Google Scholar
  11. Michael Dinitz and Zeyu Zhang. Approximating low-stretch spanners. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, 2016. Google Scholar
  12. Yevgeniy Dodis and Sanjeev Khanna. Design networks with bounded pairwise distance. In STOC '99, pages 750-759, 1999. Google Scholar
  13. Michael Elkin. Personal Communication, 2017. Google Scholar
  14. Michael Elkin and Ofer Neiman. Distributed strong diameter network decomposition: Extended abstract. In PODC '16, pages 211-216, 2016. Google Scholar
  15. Patrik Floréen, Marja Hassinen, Joel Kaasinen, Petteri Kaski, Topi Musto, and Jukka Suomela. Local approximability of max-min and min-max linear programs. Theory of Computing Systems, 2011. Google Scholar
  16. R. G. Gallager, P. A. Humblet, and P. M. Spira. A distributed algorithm for minimum-weight spanning trees. ACM Trans. Program. Lang. Syst., 5(1):66-77, 1983. Google Scholar
  17. Martin Grötschel, Lászlo Lovász, and Alexander Schrijver. Geometric Algorithms and Combinatorial Optimization, volume 2 of Algorithms and Combinatorics. Springer, 1988. Google Scholar
  18. Anupam Gupta, Mohammad T. Hajiaghayi, and Harald Räcke. Oblivious network design. In SODA '06, pages 970-979, 2006. Google Scholar
  19. M. Reza Khani and Mohammad R. Salavatipour. Improved approximations for buy-at-bulk and shallow-light k-steiner trees and (k,2)-subgraph. Journal of Combinatorial Optimization, 31(2):669-685, Feb 2016. Google Scholar
  20. Robert Krauthgamer, James R. Lee, Manor Mendel, and Assaf Naor. Measured descent: A new embedding method for finite metrics. In Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, FOCS, pages 434-443, 2004. Google Scholar
  21. Fabian Kuhn, Thomas Moscibroda, and Roger Wattenhofer. The price of being near-sighted. In SODA '06, pages 980-989, 2006. Google Scholar
  22. Nathan Linial and Michael Saks. Low diameter graph decompositions. Combinatorica, 13(4):441-454, Dec 1993. Google Scholar
  23. Christos H. Papadimitriou and Mihalis Yannakakis. Linear programming without the matrix. In STOC '93, pages 121-129, 1993. Google Scholar
  24. David Peleg. Distributed Computing: A Locality-Sensitive Approach. Society for Industrial and Applied Mathematics, 2000. Google Scholar
  25. David Peleg and Alejandro A. Schäffer. Graph spanners. Journal of Graph Theory, 13(1):99-116, 1989. Google Scholar
  26. David Peleg and Jeffrey D. Ullman. An optimal synchronizer for the hypercube. In PODC'87, pages 77-85, 1987. Google Scholar
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