Fully-Dynamic Min-Cut*

We show that we can maintain up to polylogarithmic edge connectivity for a fully-dynamic graph in \( \ifmmode\expandafter\tilde\else\expandafter\~\fi{O}{\left( {{\sqrt n }} \right)} \) worst-case time per edge insertion or deletion. Within logarithmic factors, this matches the best time bound for 1-edge connectivity. Previously, no o(n) bound was known for edge connectivity above 3, and even for 3-edge connectivity, the best update time was O(n2/3), dating back to FOCS'92.

Our algorithm maintains a concrete min-cut in terms of a pointer to a tree spanning one side of the cut plus ability to list the cut edges in O(log n) time per edge.

By dealing with polylogarithmic edge connectivity, we immediately get a sampling based expected factor (1+o(1)) approximation to general edge connectivity in \( \ifmmode\expandafter\tilde\else\expandafter\~\fi{O}{\left( {{\sqrt n }} \right)} \) time per edge insertion or deletion. This algorithm also maintains a pointer to one side of a near-minimal cut, but if we want to list the cut edges in O(log n) time per edge, the update time increases to \( \ifmmode\expandafter\tilde\else\expandafter\~\fi{O}{\left( {{\sqrt m }} \right)} \).

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Correspondence to Mikkel Thorup.

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* A preliminary version of this work was presented at the The 33rd ACM Symposium on Theory of Computing( STOC) [22], Crete, Greece, July 2001.

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Thorup, M. Fully-Dynamic Min-Cut*. Combinatorica 27, 91–127 (2007). https://doi.org/10.1007/s00493-007-0045-2

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Mathematics Subject Classification (2000):

  • 68Q25
  • 68W05
  • 68R10
  • 05C40
  • 05C85
  • 94C12
  • 94C15
  • 90B10
  • 90B25