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Parameterized Algorithms for Node Connectivity Augmentation Problems

Author Zeev Nutov

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Zeev Nutov
  • The Open University of Israel, Ra'anana, Israel

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Zeev Nutov. Parameterized Algorithms for Node Connectivity Augmentation Problems. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 92:1-92:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.92

Abstract

A graph G is k-out-connected from its node s if it contains k internally disjoint sv-paths to every node v; G is k-connected if it is k-out-connected from every node. In connectivity augmentation problems, the goal is to augment a graph G₀ = (V,E₀) by a minimum costs edge set J such that G₀ ∪ J has higher connectivity than G₀. In the k-Out-Connectivity Augmentation ({k-OCA}) problem, G₀ is (k-1)-out-connected from s and G₀ ∪ J should be k-out-connected from s; in the k-Connectivity Augmentation ({k-CA}) problem G₀ is (k-1)-connected and G₀ ∪ J should be k-connected. The parameterized complexity status of these problems was open even for k = 3 and unit costs. We will show that {k-OCA} and 3-{CA} can be solved in time 9^p ⋅ n^{O(1)}, where p is the size of an optimal solution. Our paper is the first that shows fixed-parameter tractability of a k-node-connectivity augmentation problem with high values of k. We will also consider the (2,k)-Connectivity Augmentation ({(2,k)-CA}) problem where G₀ is (k-1)-edge-connected and G₀ ∪ J should be both k-edge-connected and 2-connected. We will show that this problem can be solved in time 9^p ⋅ n^{O(1)}, and for unit costs approximated within 1.892.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • node connectivity augmentation
  • fixed-parameter tractability

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References

  1. A. Agrawal, P. Misra, F. Panolan, and S. Saurabh. Fast exact algorithms for survivable network design with uniform requirements. In WADS, pages 25-36, 2017. Google Scholar
  2. H. Angelidakis, D. H. Denesik, and L. Sanità. Node connectivity augmentation via iterative randomized rounding. CoRR, abs/2108.02041, 2021. URL: https://arxiv.org/abs/2108.02041.
  3. V. Auletta, Y. Dinitz, Z. Nutov, and D. Parente. A 2-approximation algorithm for finding an optimum 3-vertex-connected spanning subgraph. J. Algorithms, 32(1):21-30, 1999. Google Scholar
  4. J. Bang-Jensen, M. Basavaraju, K. V Klinkby, P. Misra, M. Ramanujan, S. Saurabh, and M. Zehavi. Parameterized algorithms for survivable network design with uniform demands. In SODA, pages 2838-2850, 2018. Google Scholar
  5. M. Basavaraju, F. V. Fomin, P. A. Golovach, P. Misra, M. S. Ramanujan, and S. Saurabh. Parameterized algorithms to preserve connectivity. In ICALP, Part I, pages 800-811, 2014. Google Scholar
  6. A. Belgi and Z. Nutov. A polylogarithmic approximation algorithm for 2-edge-connected dominating set. Information Processing Letters, 173:106175, 2022. Google Scholar
  7. J. Byrka, F. Grandoni, and A. J. Ameli. Breaching the 2-approximation barrier for connectivity augmentation: a reduction to Steiner tree. In Proceedings of the 52nd Annual ACM Symposium on Theory of Computing, STOC 2020, pages 815-825, 2020. Google Scholar
  8. F. Cecchetto, V. Traub, and R. Zenklusen. Bridging the gap between tree and connectivity augmentation: unified and stronger approaches. In Proceedings of the 53rd ACM Symposium on Theory of Computing, STOC 2021, pages 370-383, 2021. Google Scholar
  9. J. Cheriyan and L. A. Végh. Approximating minimum-cost k-node connected subgraphs via independence-free graphs. SIAM J. Comput., 43(4):1342-1362, 2014. Google Scholar
  10. J. Chuzhoy and S. Khanna. An O(k³log n)-approximation algorithm for vertex-connectivity survivable network design. Theory Comput., 8(1):401-413, 2012. Google Scholar
  11. E. A. Dinic, A. V. Karzanov, and M. V. Lomonosov. On the structure of a family of minimal weighted cuts in a graph. Studies in Discrete Optimization, pages 290-306, 1976. Google Scholar
  12. Y. Dinitz and Z. Nutov. A 2-level cactus model for the system of minimum and minimum+1 edge-cuts in a graph and its incremental maintenance. In Proceedings of the 27th Annual ACM Symposium on Theory of Computing, STOC 1995, pages 509-518, 1995. Google Scholar
  13. Y. Dinitz and Z. Nutov. A 3-approximation algorithm for finding optimum 4,5-vertex-connected spanning subgraphs. J. Algorithms, 32(1):31-40, 1999. Google Scholar
  14. S. E. Dreyfus and R. A. Wagner. The steiner problem in graphs. Networks, 1.3:195-207, 1971. Google Scholar
  15. A. E. Feldmann and D. Marx. The complexity landscape of fixed-parameter directed steiner network problems. In ICALP, pages 27:1-27:14, 2016. Google Scholar
  16. A. E. Feldmann, A. Mukherjee, and E.-Jan van Leeuwen. The parameterized complexity of the survivable network design problem. In SOSA, pages 37-56, 2022. Google Scholar
  17. T. Fleiner and T. Jordán. Coverings and structure of crossing families. Mathematical Programming, 84(3):505-518, 1999. Google Scholar
  18. L. Fleischer, K. Jain, and D. P. Williamson. Iterative rounding 2-approximation algorithms for minimum-cost vertex connectivity problems. Journal of Computer and System Sciences, 72(5):838-867, 2006. Google Scholar
  19. A. Frank. Rooted k-connections in digraphs. Discrete Applied Mathematics, 157(6):1242-1254, 2009. Google Scholar
  20. A. Frank and É. Tardos. An application of submodular flows. Linear Algebra Appl., 114/115:329-348, 1989. Google Scholar
  21. B. Fuchs, W. Kern, D. Molle, S. Richter, P. Rossmanith, and X. Wang. Dynamic programming for minimum steiner trees. Theory of Computing Systems, 41(3):493-500, 2007. Google Scholar
  22. J Guo and J. Uhlmann. Kernelization and complexity results for connectivity augmentation problems. Networks, 56(2):131-142, 2010. Google Scholar
  23. G. Gutin, M. Ramanujan, F. Reidl, and M. Wahlström. Path-contractions, edge deletions and connectivity preservation. Journal of Computer and System Sciences, 101:1-20, 2019. Google Scholar
  24. B. Jackson and T. Jordán. Independence free graphs and vertex connectivity augmentation. J. Comb. Theory, Ser. B, 94(1):31-77, 2005. Google Scholar
  25. K. Jain. A factor 2 approximation algorithm for the generalized steiner network problem. Combinatorica, 21(1):39-60, 2001. Google Scholar
  26. S. Khuller and B. Raghavachari. Improved approximation algorithms for uniform connectivity problems. J. Algorithms, 21(2):434-450, 1996. Google Scholar
  27. G. Kortsarz and Z. Nutov. Approximating node-connectivity problems via set covers. Algorithmica, 37:75-92, 2003. Google Scholar
  28. W. Mader. Ecken vom grad n in minimalen n-fach zusammenhängenden graphen. Archive der Mathematik, 23:219-224, 1972. Google Scholar
  29. D. Marx and L. A. Végh. Fixed-parameter algorithms for minimum-cost edge-connectivity augmentation. ACM Trans. Algorithms, 11(4):27:1-27:24, 2015. Google Scholar
  30. H. Nagamochi. An approximation for finding a smallest 2-edge-connected subgraph containing a specified spanning tree. Discrete Applied Mathematics, 126:83-113, 2003. Google Scholar
  31. Z. Nutov. Structures of Cuts and Cycles in Graphs; Algorithms and Applications. PhD thesis, Technion, Israel Institute of Technology, 1997. Google Scholar
  32. Z. Nutov. Approximating rooted connectivity augmentation problems. Algorithmica, 44(3):213-231, 2006. Google Scholar
  33. Z. Nutov. Approximating minimum-cost connectivity problems via uncrossable bifamilies. ACM Transactions on Algorithms, 9(1):1-16, 2012. Google Scholar
  34. Z. Nutov. Approximating minimum-cost edge-covers of crossing biset-families. Combinatorica, 34(1):95-114, 2014. Google Scholar
  35. Z. Nutov. Improved approximation algorithms for minimum cost node-connectivity augmentation problems. Theory Comput. Syst., 62(3):510-532, 2018. Google Scholar
  36. Z. Nutov. The k-connected subgraph problem. In T. F. Gonzalez, editor, Handbook of Approximation Algorithms and Metaheuristics, Second Edition, Volume 2: Contemporary and Emerging Applications. Chapman and Hall/CRC, 2018. Google Scholar
  37. Z. Nutov. Node-connectivity survivable network problems. In T. F. Gonzalez, editor, Handbook of Approximation Algorithms and Metaheuristics, Second Edition, Volume 2: Contemporary and Emerging Applications. Chapman and Hall/CRC, 2018. Google Scholar
  38. Z. Nutov. 2-node-connectivity network design. In WAOA, pages 220-235, 2020. Google Scholar
  39. Z. Nutov. On the tree augmentation problem. Algorithmica, 83(2):553-575, 2021. Google Scholar
  40. Z. Nutov. A 4+ε approximation for k-connected subgraphs. J. Computer and System Science, 123:64-75, 2022. Google Scholar
  41. V. Traub and R. Zenklusen. Local search for weighted tree augmentation and steiner tree. In Proceedings of ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, pages 3253-3272, 2022. Google Scholar
  42. V. Traub and R. Zenklusen. A (1.5+ε)-approximation algorithm for weighted connectivity augmentation. In Proceedings of Annual ACM Symposium on Theory of Computing, STOC 2023, pages 1820-1833, 2023. Google Scholar
  43. L. A. Végh. Augmenting undirected node-connectivity by one. SIAM J. Discret. Math, 25(2):695-718, 2011. Google Scholar
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