Faster Cut-Equivalent Trees in Simple Graphs

TL;DR

This paper presents an improved algorithm for constructing cut-equivalent trees in simple graphs, reducing the running time from subcubic to near-quadratic under certain max-flow algorithm assumptions.

Contribution

It improves the running time of constructing cut-equivalent trees from O(n^{2.5}) to O(n^2) assuming almost-linear max-flow algorithms, and to O(n^{17/8}) with the fastest current max-flow algorithms.

Findings

Improved the theoretical running time to O(n^2) with certain max-flow algorithms.

Achieved near-quadratic running time (O(n^{17/8})) using the fastest existing max-flow algorithms.

Bridged the gap between previous subcubic algorithms and practical near-linear solutions.

Abstract

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 $s, t\in V$, 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, 2021], the authors propose the first subcubic time algorithm for constructing a cut-equivalent tree. More specifically, their algorithm has $\widetilde{O}(n^{2.5})$ running time. In this paper, we improve the running time to $\hat{O}(n^2)$ if almost-linear time max-flow algorithms exist. Also, using the currently fastest max-flow algorithm by [van den Brand et al, 2021], our algorithm runs in time $\widetilde{O}(n^{17/8})$.