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Finding Small Dijoins in Transitive Closure Time

Authors Chaitanya Nalam , Thatchaphol Saranurak

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LIPIcs.FSTTCS.2025.46.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Network flows
Keywords
  • Graph algorithms
  • Dijoin
  • Submodular flow

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Abstract

We present a faster algorithm for finding a minimum dijoin, a smallest set of edges whose contraction makes a directed graph strongly connected. This problem has been studied since the 1960s [Seshu and Reed 1961] and is dual to finding a maximum sized family of disjoint dicuts [Lucchesi and Younger 1978].
Given a directed graph G with n vertices and m edges whose minimum dijoin has size d, our algorithm outputs both a minimum dijoin and a maximum sized family of disjoint dicuts in O(TC⋅ d) time, where TC = min(mn,n^ω) is the time to compute the transitive closure. This improves upon the state of the art of [Gabow 1993], which requires O(TC ⋅ min(m^{1/2},n^{2/3})) time when d = o(min(m^{1/2},n^{2/3})). Our result extends to finding a minimum weighted dijoin. We achieve this by observing that Frank’s algorithm [Frank 1981] can be sped up when warm-started with a 2-approximation solution, which we observed can be computed in near-linear time.

Cite As Get BibTex

Chaitanya Nalam and Thatchaphol Saranurak. Finding Small Dijoins in Transitive Closure Time. In 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 360, pp. 46:1-46:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.FSTTCS.2025.46

Author Details

Chaitanya Nalam
  • University of Michigan, Ann Arbor, MI, USA
Thatchaphol Saranurak
  • University of Michigan, Ann Arbor, MI, USA

Funding

  • Saranurak, Thatchaphol: Supported by NSF Grant CCF-2238138.

References

  1. Jørgen Bang-Jensen, Florian Hörsch, and Matthias Kriesell. Complexity of (arc)-connectivity problems involving arc-reversals or deorientations. Theor. Comput. Sci., 973:114097, 2023. URL: https://doi.org/10.1016/j.tcs.2023.114097.
  2. Jack Edmonds and Rick Giles. A min-max relation for submodular functions on graphs. In P.L. Hammer, E.L. Johnson, B.H. Korte, and G.L. Nemhauser, editors, Studies in Integer Programming, volume 1 of Annals of Discrete Mathematics, pages 185-204. Elsevier, 1977. URL: https://doi.org/10.1016/S0167-5060(08)70734-9.
  3. Lisa Fleischer and Satoru Iwata. Improved algorithms for submodular function minimization and submodular flow. In F. Frances Yao and Eugene M. Luks, editors, Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, May 21-23, 2000, Portland, OR, USA, pages 107-116. ACM, 2000. URL: https://doi.org/10.1145/335305.335318.
  4. András Frank. How to make a digraph strongly connected. Comb., 1(2):145-153, 1981. URL: https://doi.org/10.1007/BF02579270.
  5. András Frank. Finding feasible vectors of edmonds-giles polyhedra. Journal of Combinatorial Theory, Series B, 36(3):221-239, 1984. URL: https://doi.org/10.1016/0095-8956(84)90029-7.
  6. András Frank. Connections in combinatorial optimization. Discrete Appl. Math., page 1875, 2012. URL: https://doi.org/10.1016/j.dam.2011.09.003.
  7. András Frank and Zoltán Miklós. Simple push-relabel algorithms for matroids and submodular flows. Japan Journal of Industrial and Applied Mathematics, 29(3):419-439, 2012. URL: https://doi.org/10.1007/s13160-012-0076-y.
  8. András Frank and Éva Tardos. Generalized polymatroids and submodular flows. Mathematical Programming, 42(1):489-563, 1988. URL: https://doi.org/10.1007/BF01589418.
  9. András Frank and Éva Tardos. An application of submodular flows. Linear Algebra and its Applications, 114-115:329-348, 1989. URL: https://doi.org/10.1016/0024-3795(89)90469-2.
  10. Satoru Fujishige. Submodular functions and optimization. In Submodular Functions and Optimization, volume 58 of Annals of Discrete Mathematics, page 1. Elsevier, 1990. Google Scholar
  11. D. R. Fulkerson. Packing rooted directed cuts in a weighted directed graph. Math. Program., 6(1), 1974. URL: https://doi.org/10.1007/BF01580218.
  12. H. N. Gabow. Centroids, representations, and submodular flows. Journal of Algorithms, 18(3):586-628, 1995. URL: https://doi.org/10.1006/jagm.1995.1022.
  13. Harold N. Gabow. A representation for crossing set families with applications to submodular flow problems. In Vijaya Ramachandran, editor, Proceedings of the Fourth Annual ACM/SIGACT-SIAM Symposium on Discrete Algorithms, 25-27 January 1993, Austin, Texas, USA, pages 202-211. ACM/SIAM, 1993. URL: http://dl.acm.org/citation.cfm?id=313559.313753.
  14. H.N. Gabow. A framework for cost-scaling algorithms for submodular flow problems. In Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science, pages 449-458, 1993. URL: https://doi.org/10.1109/SFCS.1993.366842.
  15. Anupam Gupta. Lecture 2: Arborescences, 2020. Lecture Notes, CMU Course 15-850: Advanced Algorithms. Google Scholar
  16. Satoru Iwata and Yusuke Kobayashi. An algorithm for minimum cost arc-connectivity orientations. Algorithmica, 56(4):437-447, 2010. URL: https://doi.org/10.1007/s00453-008-9179-x.
  17. A. V. Karzanov. On the minimal number of arcs of a digraph meeting all its directed cutsets. Graph Theory Newsletter, 8(4), 1979. Abstract. Google Scholar
  18. Yin Tat Lee, Aaron Sidford, and Sam Chiu-wai Wong. A faster cutting plane method and its implications for combinatorial and convex optimization. CoRR, abs/1508.04874, 2015. URL: https://arxiv.org/abs/1508.04874.
  19. C. L. Lucchesi. A Minimax Equality for Directed Graphs. Ph.d. thesis, University of Waterloo, 1976. Google Scholar
  20. C. L. Lucchesi and D. H. Younger. A minimax theorem for directed graphs. Journal of the London Mathematical Society, s2-17(3):369-374, 1978. URL: https://doi.org/10.1112/jlms/s2-17.3.369.
  21. I. P. McWhirter and D. H. Younger. Strong covering of a bipartite graph. Journal of the London Mathematical Society, s2-3(1):86-90, 1971. URL: https://doi.org/10.1112/jlms/s2-3.1.86.
  22. A. Schrijver. Min-Max Results in Combinatorial Optimization, pages 439-500. Springer Berlin Heidelberg, 1983. URL: https://doi.org/10.1007/978-3-642-68874-4_18.
  23. S. Seshu and M.B. Reed. Linear Graphs and Electrical Networks. Addison-Wesley series in behavioral science: Quantitative methods. Addison-Wesley Publishing Company, 1961. URL: https://books.google.com/books?id=mMk-AAAAIAAJ.
  24. F.B. Shepherd and A. Vetta. Visualizing, finding and packing dijoins. In David Avis, Alain Hertz, and Odile Marcotte, editors, Graph Theory and Combinatorial Optimization, pages 219-254. Springer-Verlag, 2005. URL: https://doi.org/10.1007/0-387-25592-3_8.
  25. D. H. Younger. Feedback in a Directed Graph. Ph.d. thesis, Columbia University, 1963. Google Scholar
  26. U. Zimmermann. Minimization on submodular flows. Discrete Applied Mathematics, 1982. URL: https://doi.org/10.1016/0166-218X(82)90053-1.
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