Vital Edges for (s,t)-Mincut: Efficient Algorithms, Compact Structures, & Optimal Sensitivity Oracles

Authors Surender Baswana , Koustav Bhanja



PDF
Thumbnail PDF

File

LIPIcs.ICALP.2024.17.pdf
  • Filesize: 4.15 MB
  • 20 pages

Document Identifiers

Author Details

Surender Baswana
  • Department of Computer Science & Engineering, IIT Kanpur, India
Koustav Bhanja
  • Department of Computer Science & Engineering, IIT Kanpur, India

Acknowledgements

We thank Keerti Choudhary for her valuable feedback on this article.

Cite AsGet BibTex

Surender Baswana and Koustav Bhanja. Vital Edges for (s,t)-Mincut: Efficient Algorithms, Compact Structures, & Optimal Sensitivity Oracles. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 17:1-17:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.17

Abstract

Let G be a directed weighted graph on n vertices and m edges with designated source and sink vertices s and t. An edge in G is vital if its removal reduces the capacity of (s,t)-mincut. Since the seminal work of Ford and Fulkerson [CJM 1956], a long line of work has been done on computing the most vital edge and all vital edges of G. However, even after 60 years, the existing results are for either undirected or unweighted graphs. We present the following result for directed weighted graphs that also solves an open problem by Ausiello, Franciosa, Lari, and Ribichini [NETWORKS 2019]. 1. Algorithmic Results: There is an algorithm that computes all vital edges as well as the most vital edge of G using {O}(n) maximum (s,t)-flow computations. Vital edges play a crucial role in the design of sensitivity oracle for (s,t)-mincut - a compact data structure for reporting (s,t)-mincut after insertion/failure of any edge. For directed graphs, the only existing sensitivity oracle is for unweighted graphs by Picard and Queyranne [MPS 1982]. We present the first and optimal sensitivity oracle for directed weighted graphs as follows. 2. Sensitivity Oracles: a) There is an optimal O(n²) space data structure that can report an (s,t)-mincut C in O(|C|) time after the failure/insertion of any edge. b) There is an O(n) space data structure that can report the capacity of (s,t)-mincut after failure or insertion of any edge e in O(1) time if the capacity of edge e is known. A mincut for a vital edge e is an (s,t)-cut of the least capacity in which edge e is outgoing. For unweighted graphs, in a classical work, Picard and Queyranne [MPS 1982] designed an O(m) space directed acyclic graph (DAG) that stores and characterizes all mincuts for all vital edges. Conversely, there is a set containing at most n-1 (s,t)-cuts such that at least one mincut for every vital edge belongs to the set. We generalize these results for directed weighted graphs as follows. 3. Structural & Combinatorial Results: a) There is a set M containing at most n-1 (s,t)-cuts such that at least one mincut for every vital edge belongs to the set. This bound is tight as well. We also show that set M can be computed using O(n) maximum (s,t)-flow computations. b) We design two compact structures for storing and characterizing all mincuts for all vital edges - (i) an O(m) space DAG for partial and (ii) an O(mn) space structure for complete characterization. To arrive at our results, we develop new techniques, especially a generalization of maxflow-mincut Theorem by Ford and Fulkerson [CJM 1956], which might be of independent interest.

Subject Classification

ACM Subject Classification
  • Theory of computation → Network flows
  • Theory of computation → Data structures design and analysis
  • Mathematics of computing → Graph algorithms
Keywords
  • maxflow
  • vital edges
  • graph algorithms
  • structures
  • st-cuts
  • sensitivity oracle

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Ravindra K. Ahuja, Thomas L. Magnanti, and James B. Orlin. Network flows - theory, algorithms and applications. Prentice Hall, 1993. Google Scholar
  2. Yash P. Aneja, R. Chandrasekaran, and Kunhiraman Nair. Maximizing residual flow under an arc destruction. Networks, 38(4):194-198, 2001. URL: https://doi.org/10.1002/net.10001.
  3. Giorgio Ausiello, Paolo Giulio Franciosa, Isabella Lari, and Andrea Ribichini. Max flow vitality in general and st-planar graphs. Networks, 74(1):70-78, 2019. URL: https://doi.org/10.1002/net.21878.
  4. Surender Baswana, Koustav Bhanja, and Abhyuday Pandey. Minimum+1 (s, t)-cuts and dual-edge sensitivity oracle. ACM Trans. Algorithms, 19(4), October 2023. URL: https://doi.org/10.1145/3623271.
  5. Surender Baswana, Keerti Choudhary, and Liam Roditty. An efficient strongly connected components algorithm in the fault tolerant model. Algorithmica, 81:967-985, 2019. Google Scholar
  6. Surender Baswana and Abhyuday Pandey. Sensitivity oracles for all-pairs mincuts. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 581-609. SIAM, 2022. Google Scholar
  7. Michael A Bender, Martin Farach-Colton, Giridhar Pemmasani, Steven Skiena, and Pavel Sumazin. Lowest common ancestors in trees and directed acyclic graphs. Journal of Algorithms, 57(2):75-94, 2005. Google Scholar
  8. Davide Bilò, Shiri Chechik, Keerti Choudhary, Sarel Cohen, Tobias Friedrich, Simon Krogmann, and Martin Schirneck. Approximate distance sensitivity oracles in subquadratic space. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, pages 1396-1409, 2023. Google Scholar
  9. Chung-Kuan Cheng and T. C. Hu. Ancestor tree for arbitrary multi-terminal cut functions. Ann. Oper. Res., 33(3):199-213, 1991. URL: https://doi.org/10.1007/BF02115755.
  10. Keerti Choudhary. An optimal dual fault tolerant reachability oracle. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Schloss-Dagstuhl-Leibniz Zentrum für Informatik, 2016. Google Scholar
  11. Efim A Dinitz, Alexander V Karzanov, and Michael V Lomonosov. On the structure of the system of minimum edge cuts of a graph. Studies in discrete optimization, pages 290-306, 1976. Google Scholar
  12. Yefim Dinitz and Zeev 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 twenty-seventh annual ACM symposium on Theory of computing, pages 509-518, 1995. Google Scholar
  13. Yefim Dinitz and Alek Vainshtein. The general structure of edge-connectivity of a vertex subset in a graph and its incremental maintenance. odd case. SIAM Journal on Computing, 30(3):753-808, 2000. Google Scholar
  14. L. R. Ford and D. R. Fulkerson. Maximal flow through a network. Canadian Journal of Mathematics, 8:399-404, 1956. URL: https://doi.org/10.4153/CJM-1956-045-5.
  15. Greg N Frederickson and Roberto Solis-Oba. Increasing the weight of minimum spanning trees. Journal of Algorithms, 33(2):244-266, 1999. Google Scholar
  16. Loukas Georgiadis, Giuseppe F Italiano, and Nikos Parotsidis. Strong connectivity in directed graphs under failures, with applications. SIAM Journal on Computing, 49(5):865-926, 2020. Google Scholar
  17. R. E. Gomory and T. C. Hu. Multi-terminal network flows. Journal of the Society for Industrial and Applied Mathematics, 9(4):551-570, 1961. URL: http://www.jstor.org/stable/2098881.
  18. Ralph E Gomory and Tien Chung Hu. Multi-terminal network flows. Journal of the Society for Industrial and Applied Mathematics, 9(4):551-570, 1961. Google Scholar
  19. Fabrizio Grandoni and Virginia Vassilevska Williams. Faster replacement paths and distance sensitivity oracles. ACM Trans. Algorithms, 16(1):15:1-15:25, 2020. URL: https://doi.org/10.1145/3365835.
  20. Frieda Granot and Refael Hassin. Multi-terminal maximum flows in node-capacitated networks. Discret. Appl. Math., 13(2-3):157-163, 1986. URL: https://doi.org/10.1016/0166-218X(86)90079-X.
  21. Dan Gusfield and Dalit Naor. Efficient algorithms for generalized cut-trees. Networks, 21(5):505-520, 1991. Google Scholar
  22. Refael Hassin and Asaf Levin. Flow trees for vertex-capacitated networks. Discrete applied mathematics, 155(4):572-578, 2007. Google Scholar
  23. John Hershberger and Subhash Suri. Erratum to "Vickrey pricing and shortest paths: What is an edge worth?". In Annual Symposium on Foundation of Computer Science, volume 43, pages 809-810. IEEE Computer Society Press, 2002. Google Scholar
  24. Giuseppe F Italiano, Adam Karczmarz, and Nikos Parotsidis. Planar reachability under single vertex or edge failures. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2739-2758. SIAM, 2021. Google Scholar
  25. Michal Katz, Nir A. Katz, Amos Korman, and David Peleg. Labeling schemes for flow and connectivity. SIAM J. Comput., 34(1):23-40, 2004. URL: https://doi.org/10.1137/S0097539703433912.
  26. Kao-Chêng Lin and Maw-Sheng Chern. The most vital edges in the minimum spanning tree problem. Inf. Process. Lett., 45(1):25-31, 1993. URL: https://doi.org/10.1016/0020-0190(93)90247-7.
  27. Stephen H Lubore, HD Ratliff, and GT Sicilia. Determining the most vital link in a flow network. Naval Research Logistics Quarterly, 18(4):497-502, 1971. Google Scholar
  28. Enrico Nardelli, Guido Proietti, and Peter Widmayer. A faster computation of the most vital edge of a shortest path. Information Processing Letters, 79(2):81-85, 2001. Google Scholar
  29. Cynthia A Phillips. The network inhibition problem. In Proceedings of the twenty-fifth annual ACM symposium on Theory of computing, pages 776-785, 1993. Google Scholar
  30. Jean-Claude Picard and Maurice Queyranne. On the structure of all minimum cuts in a network and applications. In Rayward-Smith V.J. (eds) Combinatorial Optimization II. Mathematical Programming Studies, 13(1):8-16, 1980. URL: https://doi.org/10.1007/BFb0120902.
  31. H Donald Ratliff, G Thomas Sicilia, and SH Lubore. Finding the n most vital links in flow networks. Management Science, 21(5):531-539, 1975. Google Scholar
  32. Liam Roditty and Uri Zwick. Replacement paths and k simple shortest paths in unweighted directed graphs. ACM Transactions on Algorithms (TALG), 8(4):1-11, 2012. Google Scholar
  33. Fu-Shang P Tsen, Ting-Yi Sung, Men-Yang Lin, Lih-Hsing Hsu, and Wendy Myrvold. Finding the most vital edges with respect to the number of spanning trees. IEEE Transactions on Reliability, 43(4):600-603, 1994. Google Scholar
  34. Virginia Vassilevska Williams, Eyob Woldeghebriel, and Yinzhan Xu. Algorithms and lower bounds for replacement paths under multiple edge failure. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 907-918. IEEE, 2022. Google Scholar
  35. Richard D Wollmer. Some methods for determining the most vital link in a railway network. Rand Corporation, 1963. Google Scholar