Faster Cut-Equivalent Trees in Simple Graphs

Author Tianyi Zhang

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Tianyi Zhang
  • Tel Aviv University, Israel


The author would like to thank helpful discussions with Prof. Ran Duan and Dr. Amir Abboud.

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Tianyi Zhang. Faster Cut-Equivalent Trees in Simple Graphs. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 109:1-109:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


Let G = (V, E) be an undirected connected simple graph on n vertices. A cut-equivalent tree of G is an edge-weighted tree on the same vertex set V, such that for any pair of vertices s, t ∈ V, the minimum (s, t)-cut in the tree is also a minimum (s, t)-cut in G, and these two cuts have the same cut value. In a recent paper [Abboud, Krauthgamer and Trabelsi, STOC 2021], the authors propose the first subcubic time algorithm for constructing a cut-equivalent tree. More specifically, their algorithm has Õ(n^{2.5}) running time. Later on, this running time was significantly improved to n^{2+o(1)} by two independent works [Abboud, Krauthgamer and Trabelsi, FOCS 2021] and [Li, Panigrahi, Saranurak, FOCS 2021], and then to (m+n^{1.9})^{1+o(1)} by [Abboud, Krauthgamer and Trabelsi, SODA 2022]. In this paper, we improve the running time to Õ(n²) graphs if near-linear time max-flow algorithms exist, or Õ(n^{17/8}) using the currently fastest max-flow algorithm. Although our algorithm is slower than previous works, the runtime bound becomes better by a sub-polynomial factor in dense simple graphs when assuming near-linear time max-flow algorithms.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • graph algorithms
  • minimum cuts
  • max-flow


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