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DOI: 10.4230/LIPIcs.FSTTCS.2025.46

Finding Small Dijoins in Transitive Closure Time

Authors: Chaitanya Nalam and Thatchaphol Saranurak

Published in: LIPIcs, Volume 360, 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)

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.

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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)

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@InProceedings{nalam_et_al:LIPIcs.FSTTCS.2025.46,
  author =	{Nalam, Chaitanya and Saranurak, Thatchaphol},
  title =	{{Finding Small Dijoins in Transitive Closure Time}},
  booktitle =	{45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)},
  pages =	{46:1--46:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-406-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{360},
  editor =	{Aiswarya, C. and Mehta, Ruta and Roy, Subhajit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2025.46},
  URN =		{urn:nbn:de:0030-drops-251265},
  doi =		{10.4230/LIPIcs.FSTTCS.2025.46},
  annote =	{Keywords: Graph algorithms, Dijoin, Submodular flow}
}

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Finding Small Dijoins in Transitive Closure Time

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