Cactus Representation of Minimum Cuts: Derandomize and Speed up

TL;DR

This work settles the long-standing open question of near-linear deterministic computation of the cactus representation for all global mincuts in undirected weighted graphs. By building a small tree packing and assigning compact cut labels to minimal 2-respecting mincuts, it reduces cactus construction to an efficient aggregation over trees using path decompositions and dynamic-tree data structures. The authors introduce a complete deterministic algorithm for minimal 2-respecting mincuts with both comparable and incomparable cases, supported by new structural lemmas and a full proof for cactus construction. The approach yields a deterministic $m^{1+o(1)}$-time algorithm and improves the best randomized bound to $O(m\log^3 n)$, with immediate applications to $+1$-edge-connectivity augmentation and related problems.

Abstract

Given an undirected weighted graph with $n$ vertices and $m$ edges, we give the first deterministic $m^{1+o(1)}$-time algorithm for constructing the cactus representation of \emph{all} global minimum cuts. This improves the current $n^{2+o(1)}$-time state-of-the-art deterministic algorithm, which can be obtained by combining ideas implicitly from three papers [Karger JACM'2000, Li STOC'2021, and Gabow TALG'2016] The known explicitly stated deterministic algorithm has a runtime of $\tilde{O}(mn)$ [Fleischer 1999, Nagamochi and Nakao 2000]. Using our technique, we can even speed up the fastest randomized algorithm of [Karger and Panigrahi, SODA'2009] whose running time is at least $Ω(m\log^4 n)$ to $O(m\log^3 n)$.

Figures (5)

• Figure 1: High level idea: once the lower vertex of an edge $l_e$ has found, we may locate the upper vertex $u_e$ along the path via a $\texttt{MinPath}^\downarrow$ query.
• Figure 2: Querying $x_e$ must find $l_e$. Querying at other vertices might not find $l_e$.
• Figure 3: Simple Case: $l_e$ can be found by $\texttt{MinNonPath}\xspace(u_1, u_2)$ after separating the subtree rooted at $u=\textsc{lca}_e$.
• Figure 4: An illustration to \ref{['lem:general-case-property']}: if the algorithm arrives at vertex $v$ but has not found any lower vertex for $e_1$ and $e_2$ yet, then $\hat{P}_{e_1}=\hat{P}_{e_2}$.
• Figure 5: A Missing Case.

Theorems & Definitions (79)

• theorem 1.1
• definition 2.1: Minimal mincuts
• lemma 2.2: dinits1976structure
• lemma 2.3: karger2009near
• theorem 3.1
• lemma 3.1
• lemma 3.2
• proof
• lemma 3.3: karger2009near
• lemma 3.4: Key Lemma