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Degrees and Network Design: New Problems and Approximations

Authors Michael Dinitz, Guy Kortsarz, Shi Li

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Author Details

Michael Dinitz
  • Johns Hopkins University, Baltimore, MD, USA
Guy Kortsarz
  • Rutgers University, Camden, NJ, USA
Shi Li
  • Nanjing University, Jiangsu, China

Funding

  • Dinitz, Michael: Supported in part by NSF awards CCF-1909111 and CCF-2228995.

Cite AsGet BibTex

Michael Dinitz, Guy Kortsarz, and Shi Li. Degrees and Network Design: New Problems and Approximations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 3:1-3:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.3

Abstract

While much of network design focuses mostly on cost (number or weight of edges), node degrees have also played an important role. They have traditionally either appeared as an objective, to minimize the maximum degree (e.g., the Minimum Degree Spanning Tree problem), or as constraints that might be violated to give bicriteria approximations (e.g., the Minimum Cost Degree Bounded Spanning Tree problem). We extend the study of degrees in network design in two ways. First, we introduce and study a new variant of the Survivable Network Design Problem where in addition to the traditional objective of minimizing the cost of the chosen edges, we add a constraint that the 𝓁_p-norm of the node degree vector is bounded by an input parameter. This interpolates between the classical settings of maximum degree (the 𝓁_∞-norm) and the number of edges (the 𝓁₁-degree), and has natural applications in distributed systems and VLSI design. We give a constant bicriteria approximation in both measures using convex programming. Second, we provide a polylogarithmic bicriteria approximation for the Degree Bounded Group Steiner problem on bounded treewidth graphs, solving an open problem from [Guy Kortsarz and Zeev Nutov, 2022] and [X. Guo et al., 2022].

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Routing and network design problems
Keywords
  • Network Design
  • Degrees

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References

  1. B. Awerbuch, Y. Azar, E. Grove, M. Kao, P. Krishnan, and J. Vitter. Load balancing in the l_p norm. In FOCS, pages 383-391, 1995. Google Scholar
  2. Y. Azar and S. Taub. All-norm approximation for scheduling on identical machines. In T. Hagerup and J. Katajainen, editors, SWAT, volume 3111, pages 298-310, 2004. Google Scholar
  3. Hans L. Bodlaender. Nc-algorithms for graphs with small treewidth. In Jan van Leeuwen, editor, Graph-Theoretic Concepts in Computer Science, 14th International Workshop, WG '88, Amsterdam, The Netherlands, June 15-17, 1988, Proceedings, volume 344 of Lecture Notes in Computer Science, pages 1-10. Springer, 1988. URL: https://doi.org/10.1007/3-540-50728-0_32.
  4. P. Chalermsook, S. Das, B. Laekhanukit, and D. Vaz. Beyond metric embedding: Approximating group steiner trees on bounded treewidth graphs. In SODA, pages 737-751, 2017. Google Scholar
  5. Eden Chlamtác, Michael Dinitz, and Thomas Robinson. Approximating the Norms of Graph Spanners. In Dimitris Achlioptas and László A. Végh, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019), volume 145 of Leibniz International Proceedings in Informatics (LIPIcs), pages 11:1-11:22, Dagstuhl, Germany, 2019. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.11.
  6. Eden Chlamtác, Michael Dinitz, and Thomas Robinson. The Norms of Graph Spanners. In Christel Baier, Ioannis Chatzigiannakis, Paola Flocchini, and Stefano Leonardi, editors, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019), volume 132 of Leibniz International Proceedings in Informatics (LIPIcs), pages 40:1-40:15, Dagstuhl, Germany, 2019. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.40.
  7. Irit Dinur and David Steurer. Analytical approach to parallel repetition. In STOC, pages 624-633, 2014. Google Scholar
  8. M. Furer and B. Raghavachari. Approximating the minimum-degree steiner tree to within one of optimal. Journal of Algorithms, 17(3):409-423, 1994. Google Scholar
  9. N. Garg, G. Konjevod, and R. Ravi. A polylogarithmic approximation algorithm for the group Steiner tree problem. J. Algorithms, 37(1):66-84, 2000. Google Scholar
  10. D. Golovin, A. Gupta, A. Kumar, and K. Tangwongsan. All-norms and all-l_p-norms approximation algorithms. In IACS, volume 2 of LIPIcs, pages 199-210, 2008. Google Scholar
  11. Fabrizio Grandoni, Bundit Laekhanukit, and Shi Li. o(log²k/log logk)-approximation algorithm for directed steiner tree: A tight quasi-polynomial time algorithm. SIAM Journal on Computing, 52(2):STOC19-298-STOC19-322, 2023. URL: https://doi.org/10.1137/20M1312988.
  12. X. Guo, G. Kortsarz, B. Laekhanukit, S. Li, D. Vaz, and J. Xian. On approximating degree-bounded network design problems. Algorithmica, 84(5):1252-1278, 2022. Google Scholar
  13. Mohammad Taghi Hajiaghayi. A list of open problems in bounded degree network design. The 8'th Workshop on Flexible Network Design, 2016. Google Scholar
  14. E. Halperin and R. Krauthgamer. Polylogarithmic inapproximability. In STOC, pages 585-594, 2003. Google Scholar
  15. Sungjin Im and Shi Li. Improved approximations for unrelated machine scheduling. In Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2917-2946, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch111.
  16. K. Jain. A factor 2 approximation algorithm for the generalized steiner network problem. Combinatorica, 21(1):39-60, 2001. Google Scholar
  17. Guy Kortsarz and Zeev Nutov. The minimum degree group steiner problem. Discret. Appl. Math., 309:229-239, 2022. Google Scholar
  18. V. Kumar, M. Marathe, S. Parthasarathy, and A. Srinivasan. A unified approach to scheduling on unrelated parallel machines. J. ACM, 56(5):28:1-28:31, 2009. Google Scholar
  19. C. Lau L, R. Ravi, and M. Singh. Iterative Methods in Combinatorial Optimization. Cambridge University Press, 2011. Google Scholar
  20. L. C. Lau, J. Naor, M. R. Salavatipour, and M. Singh. Survivable network design with degree or order constraints. SIAM J. Comput., 39(3):1062-1087, 2009. Google Scholar
  21. Shi Li, Chenyang Xu, and Ruilong Zhang. Polylogarithmic Approximation for Robust s-t Path. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024), 2024. Google Scholar
  22. Manmeet Kaur Nisha Sharma. A survey of vlsi techniques for power optimization and estimation of optimization. International Journal of Emerging Technology and Advanced Engineering, 4, 2014. Google Scholar
  23. Mohit Singh and Lap Chi Lau. Approximating minimum bounded degree spanning trees to within one of optimal. J. ACM, 62(1), March 2015. URL: https://doi.org/10.1145/2629366.
  24. Y. Wang, X. Hong, T. Jing, Y. Yang, X. Hu, and Guiying Yan. An efficient low-degree RMST algorithm for VLSI/ULSI physical design. In PATMOS, pages 442-452, 2004. Google Scholar
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