Faster Algorithms for All-Pairs Bounded Min-Cuts

Authors Amir Abboud, Loukas Georgiadis, Giuseppe F. Italiano, Robert Krauthgamer, Nikos Parotsidis, Ohad Trabelsi, Przemysław Uznański, Daniel Wolleb-Graf

Thumbnail PDF


  • Filesize: 0.61 MB
  • 15 pages

Document Identifiers

Author Details

Amir Abboud
  • IBM Almaden Research Center, California, USA
Loukas Georgiadis
  • University of Ioannina, Greece
Giuseppe F. Italiano
  • LUISS University, Rome, Italy
Robert Krauthgamer
  • Weizmann Institute of Science, Israel
Nikos Parotsidis
  • University of Copenhagen, Denmark
Ohad Trabelsi
  • Weizmann Institute of Science, Israel
Przemysław Uznański
  • University of Wrocław, Poland
Daniel Wolleb-Graf
  • ETH Zürich, Switzerland


We thank Paweł Gawrychowski, Mohsen Ghaffari, Atri Rudra and Peter Widmayer for the valuable discussions on this problem.

Cite AsGet BibTex

Amir Abboud, Loukas Georgiadis, Giuseppe F. Italiano, Robert Krauthgamer, Nikos Parotsidis, Ohad Trabelsi, Przemysław Uznański, and Daniel Wolleb-Graf. Faster Algorithms for All-Pairs Bounded Min-Cuts. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


The All-Pairs Min-Cut problem (aka All-Pairs Max-Flow) asks to compute a minimum s-t cut (or just its value) for all pairs of vertices s,t. We study this problem in directed graphs with unit edge/vertex capacities (corresponding to edge/vertex connectivity). Our focus is on the k-bounded case, where the algorithm has to find all pairs with min-cut value less than k, and report only those. The most basic case k=1 is the Transitive Closure (TC) problem, which can be solved in graphs with n vertices and m edges in time O(mn) combinatorially, and in time O(n^{omega}) where omega<2.38 is the matrix-multiplication exponent. These time bounds are conjectured to be optimal. We present new algorithms and conditional lower bounds that advance the frontier for larger k, as follows: - A randomized algorithm for vertex capacities that runs in time {O}((nk)^{omega}). This is only a factor k^omega away from the TC bound, and nearly matches it for all k=n^{o(1)}. - Two deterministic algorithms for edge capacities (which is more general) that work in DAGs and further reports a minimum cut for each pair. The first algorithm is combinatorial (does not involve matrix multiplication) and runs in time {O}(2^{{O}(k^2)}* mn). The second algorithm can be faster on dense DAGs and runs in time {O}((k log n)^{4^{k+o(k)}}* n^{omega}). Previously, Georgiadis et al. [ICALP 2017], could match the TC bound (up to n^{o(1)} factors) only when k=2, and now our two algorithms match it for all k=o(sqrt{log n}) and k=o(log log n). - The first super-cubic lower bound of n^{omega-1-o(1)} k^2 time under the 4-Clique conjecture, which holds even in the simplest case of DAGs with unit vertex capacities. It improves on the previous (SETH-based) lower bounds even in the unbounded setting k=n. For combinatorial algorithms, our reduction implies an n^{2-o(1)} k^2 conditional lower bound. Thus, we identify new settings where the complexity of the problem is (conditionally) higher than that of TC. Our three sets of results are obtained via different techniques. The first one adapts the network coding method of Cheung, Lau, and Leung [SICOMP 2013] to vertex-capacitated digraphs. The second set exploits new insights on the structure of latest cuts together with suitable algebraic tools. The lower bounds arise from a novel reduction of a different structure than the SETH-based constructions.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Network flows
  • All-pairs min-cut
  • k-reachability
  • network coding
  • Directed graphs
  • fine-grained complexity


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


  1. A. Abboud and V. V. Williams. Popular Conjectures Imply Strong Lower Bounds for Dynamic Problems. In FOCS, pages 434-443, October 2014. URL:
  2. Amir Abboud, Arturs Backurs, Karl Bringmann, and Marvin Künnemann. Fine-Grained Complexity of Analyzing Compressed Data: Quantifying Improvements over Decompress-and-Solve. In FOCS, pages 192-203, 2017. URL:
  3. Amir Abboud, Arturs Backurs, and Virginia Vassilevska Williams. If the Current Clique Algorithms are Optimal, So is Valiant’s Parser. In FOCS, pages 98-117, 2015. URL:
  4. Amir Abboud, Robert Krauthgamer, and Ohad Trabelsi. New Algorithms and Lower Bounds for All-Pairs Max-Flow in Undirected Graphs. CoRR, abs/1901.01412, 2019. URL:
  5. Amir Abboud, Virginia Vassilevska Williams, and Huacheng Yu. Matching Triangles and Basing Hardness on an Extremely Popular Conjecture. SIAM J. Comput., 47(3):1098-1122, 2018. URL:
  6. Srinivasa R Arikati, Shiva Chaudhuri, and Christos D Zaroliagis. All-Pairs Min-Cut in Sparse Networks. Journal of Algorithms, 29(1):82-110, 1998. URL:
  7. Attila Bernáth and Gyula Pap. Blocking unions of arborescences. CoRR, abs/1507.00868, 2015. URL:
  8. Anand Bhalgat, Ramesh Hariharan, Telikepalli Kavitha, and Debmalya Panigrahi. An Õ(mn) Gomory-Hu Tree Construction Algorithm for Unweighted Graphs. In STOC, pages 605-614, 2007. URL:
  9. Karl Bringmann, Allan Grønlund, and Kasper Green Larsen. A Dichotomy for Regular Expression Membership Testing. In FOCS, pages 307-318, 2017. URL:
  10. Karl Bringmann and Philip Wellnitz. Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars. In CPM, pages 12:1-12:14, 2017. URL:
  11. Timothy M. Chan. All-Pairs Shortest Paths with Real Weights in O(n³/log n) Time. Algorithmica, 50(2):236-243, 2008. URL:
  12. Yi-Jun Chang. Conditional Lower Bound for RNA Folding Problem. CoRR, abs/1511.04731, 2015. URL:
  13. Ho Yee Cheung, Lap Chi Lau, and Kai Man Leung. Graph Connectivities, Network Coding, and Expander Graphs. SIAM Journal on Computing, 42(3):733-751, 2013. URL:
  14. D. Coppersmith and S. Winograd. Matrix multiplication via arithmetic progressions. Journal of Symbolic Computation, 9(3):251-280, 1990. URL:
  15. Friedrich Eisenbrand and Fabrizio Grandoni. On the complexity of fixed parameter clique and dominating set. Theor. Comput. Sci., 326(1-3):57-67, 2004. URL:
  16. M. J. Fischer and A. R. Meyer. Boolean matrix multiplication and transitive closure. In SWAT, pages 129-131. IEEE, 1971. URL:
  17. Lester Randolph Ford, Jr. and Delbert Ray Fulkerson. Flows in Networks. Princeton University Press, 1962. Google Scholar
  18. Loukas Georgiadis, Daniel Graf, Giuseppe F. Italiano, Nikos Parotsidis, and Przemysław Uznański. All-Pairs 2-Reachability in O(n^ω log n) Time. In ICALP, volume 80, pages 74:1-74:14, 2017. URL:
  19. 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:
  20. Ramesh Hariharan, Telikepalli Kavitha, and Debmalya Panigrahi. Efficient Algorithms for Computing All Low s-t Edge Connectivities and Related Problems. In SODA, pages 127-136, 2007. URL:
  21. Robert Krauthgamer and Ohad Trabelsi. Conditional Lower Bounds for All-Pairs Max-Flow. ACM Trans. Algorithms, 14(4):42:1-42:15, 2018. URL:
  22. Jakub Łącki, Yahav Nussbaum, Piotr Sankowski, and Christian Wulff-Nilsen. Single Source - All Sinks Max Flows in Planar Digraphs. In FOCS, pages 599-608. IEEE Computer Society, 2012. URL:
  23. F. Le Gall. Powers of Tensors and Fast Matrix Multiplication. In ISSAC, pages 296-303, 2014. URL:
  24. Y. T. Lee and A. Sidford. Path Finding Methods for Linear Programming: Solving Linear Programs in Õ(√rank) Iterations and Faster Algorithms for Maximum Flow. In FOCS, pages 424-433, 2014. URL:
  25. Andrea Lincoln, Virginia Vassilevska Williams, and R. Ryan Williams. Tight Hardness for Shortest Cycles and Paths in Sparse Graphs. In SODA, pages 1236-1252, 2018. Google Scholar
  26. A. Mądry. Computing Maximum Flow with Augmenting Electrical Flows. In FOCS, pages 593-602, 2016. URL:
  27. Dániel Marx. Parameterized graph separation problems. Theoretical Computer Science, 351(3):394-406, 2006. Google Scholar
  28. K. Menger. Zur allgemeinen Kurventheorie. Fundamenta Mathematicae, 10(1):96-115, 1927. Google Scholar
  29. Hiroshi Nagamochi and Yoko Kamidoi. Minimum cost subpartitions in graphs. Inf. Process. Lett., 102(2-3):79-84, 2007. URL:
  30. Jaroslav Nešetřil and Svatopluk Poljak. On the complexity of the subgraph problem. Commentationes Mathematicae Universitatis Carolinae, 26(2):415-419, 1985. Google Scholar
  31. Debmalya Panigrahi. Gomory-Hu trees. In Ming-Yang Kao, editor, Encyclopedia of Algorithms, pages 858-861. Springer, 2016. URL:
  32. Virginia Vassilevska. Efficient algorithms for clique problems. Inf. Process. Lett., 109(4):254-257, 2009. URL:
  33. V. Vassilevska Williams. Multiplying Matrices Faster Than Coppersmith-Winograd. In STOC, pages 887-898, 2012. URL:
  34. Virginia Vassilevska Williams and R. Ryan Williams. Subcubic Equivalences Between Path, Matrix, and Triangle Problems. J. ACM, 65(5):27:1-27:38, 2018. URL:
  35. Huacheng Yu. An improved combinatorial algorithm for Boolean matrix multiplication. Inf. Comput., 261:240-247, 2018. URL:
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail