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Approximating k-Connected m-Dominating Sets

Author Zeev Nutov

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Zeev Nutov
  • The Open University of Israel, Raanana, Israel

Acknowledgements

I thank an anonymous referee for many useful comments.

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Zeev Nutov. Approximating k-Connected m-Dominating Sets. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 73:1-73:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ESA.2020.73

Abstract

A subset S of nodes in a graph G is a k-connected m-dominating set ((k,m)-cds) if the subgraph G[S] induced by S is k-connected and every v ∈ V⧵S has at least m neighbors in S. In the k-Connected m-Dominating Set ((k,m)-CDS) problem the goal is to find a minimum weight (k,m)-cds in a node-weighted graph. For m ≥ k we obtain the following approximation ratios. For general graphs our ratio O(k ln n) improves the previous best ratio O(k² ln n) of [Z. Nutov, 2018] and matches the best known ratio for unit weights of [Z. Zhang et al., 2018]. For unit disk graphs we improve the ratio O(k ln k) of [Z. Nutov, 2018] to min{m/(m-k),k^{2/3}} ⋅ O(ln² k) - this is the first sublinear ratio for the problem, and the first polylogarithmic ratio O(ln² k)/ε when m ≥ (1+ε)k; furthermore, we obtain ratio min{m/(m-k), √k} ⋅ O(ln² k) for uniform weights. These results are obtained by showing the same ratios for the Subset k-Connectivity problem when the set of terminals is an m-dominating set.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • k-connected graph
  • m-dominating set
  • approximation algorithm
  • rooted subset k-connectivity
  • subset k-connectivity

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References

  1. C. Ambühl, T. Erlebach, M. Mihalák, and M. Nunkesser. Constant-factor approximation for minimum weight (connected) dominating sets in unit disk graphs. In APPROX-RANDOM, pages 3-14, 2006. Google Scholar
  2. A. Belgi and Z. Nutov. An Õ (log² n)-approximation algorithm for 2-edge-connected dominating set. CoRR abs/1912.09662, 2019. URL: http://arxiv.org/abs/1912.09662.
  3. X. Cheng, X. Huang, D. Li, W. Weili, and D.-Z. Du. A polynomial-time approximation scheme for the minimum-connected dominating set in ad hoc wireless networks. Networks, 42(4):202-208, 2003. Google Scholar
  4. J. Cheriyan and L Végh. Approximating minimum-cost k-node connected subgraphs via independence-free graphs. SIAM J. Comput., 43(4):1342-1362, 2014. Preliminary version in FOCS 2013. Google Scholar
  5. B. N. Clark, C. J. Colbourn, and D. S. Johnson. Unit disk graphs. Discrete Mathematics, 86(1-3):165-177, 1990. Google Scholar
  6. B. Das and V. Bharghavan. Routing in ad-hoc networks using minimum connected dominating sets. In IEEE International Conference on Comunications, Montreal, page 376–380, 1997. Google Scholar
  7. A. Ephremides, J. E. Wieselthier, and D. J. Baker. A design concept for reliable mobile radio networks with frequency hopping signaling. Proceedings of the IEEE, 75(1):56-73, 1987. Google Scholar
  8. K.-T. Förster. Approximating fault-tolerant domination in general graphs. In ANALCO, pages 25-32, 2013. Google Scholar
  9. T. Fukunaga. Spider covers for prize-collecting network activation problem. ACM Trans. Algorithms, 13(4):49:1-49:31, 2017. preliminary version in SODA 2015. Google Scholar
  10. T. Fukunaga. Approximation algorithms for highly connected multi-dominating sets in unit disk graphs. Algorithmica, 80(11):3270-3292, 2018. Google Scholar
  11. M. Goemans, A. Goldberg, S. Plotkin, D. Shmoys, E. Tardos, and D. Williamson. Improved approximation algorithms for network design problems. In SODA, pages 223-232, 1994. Google Scholar
  12. S. Guha and S. Khuller. Approximation algorithms for connected dominating sets. Algorithmica, 20(4):374-387, 1998. preliminary version in ESA 1996. Google Scholar
  13. S. Guha and S. Khuller. Improved methods for approximating node weighted steiner trees and connected dominating sets. Inf. Comput., 150(1):57-74, 1999. Google Scholar
  14. B. Jackson and T. Jordán. Independence free graphs and vertex connectivity augmentation. J. Comb. Theory, Ser. B, 94(1):31-77, 2005. Preliminary version in IPCO 2001. Google Scholar
  15. T. Jordán. On the optimal vertex-connectivity augmentation. J. Combin. Theory Ser. B, 63:8-20, 1995. Google Scholar
  16. P. Klein and R. Ravi. A nearly best-possible approximation algorithm for node-weighted steiner trees. J. Algorithms, 19(1):104-115, 1995. Preliminary version in IPCO 1993. Google Scholar
  17. G. Kortsarz and Z. Nutov. Approximating node connectivity problems via set covers. Algorithmica, 37:75-92, 2003. Preliminary version in APPROX 2000. Google Scholar
  18. G. Kortsarz and Z. Nutov. Approximating source location and star survivable network problems. Theoretical Computer Science, 674:32-42, 2017. Google Scholar
  19. B. Laekhanukit. An improved approximation algorithm for the minimum cost subset k-connected subgraph problem. Algorithmica, 72(3):714-733, 2015. Preliminary version in ICALP 2011. Google Scholar
  20. W. Mader. Ecken vom grad n in minimalen n-fach zusammenhängenden graphen. Archive der Mathematik, 23:219-224, 1972. Google Scholar
  21. M. V. Marathe, H. Breu, H. B. Hunt III, S. S. Ravi, and D. J. Rosenkrantz. Simple heuristics for unit disk graphs. Networks, 25(2):59-68, 1995. Google Scholar
  22. Z. Nutov. Approximating minimum cost connectivity problems via uncrossable bifamilies. ACM Transactions on Algorithms, 9(1):1:1-1:16, 2012. Preliminary version in FOCS 2009. Google Scholar
  23. Z. Nutov. Approximating subset k-connectivity problems. J. Discrete Algorithms, 17:51-59, 2012. Google Scholar
  24. Z. Nutov. Activation network design problems. In T. F. Gonzalez, editor, Handbook on Approximation Algorithms and Metaheuristics, Second Edition, volume 2, chapter 12. Chapman & Hall/CRC, 2018. Google Scholar
  25. Z. Nutov. Erratum: Approximating minimum cost connectivity problems via uncrossable bifamilies. ACM Transactions on Algorithms, 14(3):37:1-37:8, 2018. Google Scholar
  26. Z. Nutov. Improved approximation algorithms for k-connected m-dominating set problems. Information Processing Letters, 140:30-33, 2018. Google Scholar
  27. Z. Nutov. The k-connected subgraph problem. In T. F. Gonzalez, editor, Handbook on Approximation Algorithms and Metaheuristics, Second Edition, volume 2, chapter 15. Chapman & Hall/CRC, 2018. Google Scholar
  28. Z. Nutov. Node-connectivity survivable network problems. In T. F. Gonzalez, editor, Handbook on Approximation Algorithms and Metaheuristics, Second Edition, volume 2, chapter 13. Chapman & Hall/CRC, 2018. Google Scholar
  29. Z. Nutov. 2-node-connectivity network design. CoRR abs/2002.04048, 2020. URL: http://arxiv.org/abs/2002.04048.
  30. Z. Nutov. A 4+ε approximation for k-connected subgraphs. In SODA, pages 1000-1009, 2020. Google Scholar
  31. Y. Shi, Z. Zhang, Y. Mo, and D.-Z. Du. Approximation algorithm for minimum weight fault-tolerant virtual backbone in unit disk graphs. IEEE/ACM Transactions on networking, 25(2):925-933, 2017. Google Scholar
  32. M. Thai, N. Zhang, R. Tiwari, and X. Xu. On approximation algorithms of k-connected m-dominating sets in disk graphs. Theoretical Computer Science, 385:49-59, 2007. Google Scholar
  33. J. Willson, Z. Zhang, W. Wu, and D.-Z. Du. Fault-tolerant coverage with maximum lifetime in wireless sensor networks. In INFOCOM, pages 1364-1372, 2015. Google Scholar
  34. Z. Zhang, J. Zhou, S. Tang, X. Huang, and D.-Z. Du. Computing minimum k-connected m-fold dominating set in general graphs. INFORMS Journal on Computing, 30(2):217-224, 2018. Google Scholar
  35. F. Zou, X. Li, S. Gao, and W. Weili. Node-weighted steiner tree approximation in unit disk graphs. J. Combinatorial Optimization, 18(4):342-349, 2009. Google Scholar
  36. F. Zou, Y. Wang, X. XiaoHua, X. Li, D. Hongwei, P.-J. Wan, and W. Weili. New approximations for minimum-weighted dominating sets and minimum-weighted connected dominating sets on unit disk graphs. Theoretical Computer Science, 412(3):198-208, 2011. Google Scholar
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