Deterministic Near-Linear Time Minimum Cut in Weighted Graphs

TL;DR

The paper presents a deterministic near-linear time algorithm for computing minimum cuts in weighted graphs by introducing a sparse clustering that preserves minimum cuts with an $o(1)$ multiplicative error and combining this with a skeleton-graph framework. It builds on and extends Li21’s skeleton approach, Kawarabayashi–Thorup’s decomposition ideas, and weighted-graph flow techniques to handle crossing cuts and recursion depth, achieving near-linear performance without randomness. The core toolkit includes an adapted Unit-Flow procedure for weighted graphs, a scalable $s_0$-strong partitioning method, efficient uncrossing for small and large clusters, and a two-pronged sparsifier construction (unbalanced and balanced) augmented with Steiner vertices and cutting/splitting steps. The resulting deterministic algorithm yields a near-linear time minimum-cut computation with provable guarantees, paving the way for robust, scalable minimum-cut analysis on weighted networks with practical impact in graph algorithms and network design.

Abstract

In 1996, Karger [Kar96] gave a startling randomized algorithm that finds a minimum-cut in a (weighted) graph in time $O(m\log^3n)$ which he termed near-linear time meaning linear (in the size of the input) times a polylogarthmic factor. In this paper, we give the first deterministic algorithm which runs in near-linear time for weighted graphs. Previously, the breakthrough results of Kawarabayashi and Thorup [KT19] gave a near-linear time algorithm for simple graphs. The main technique here is a clustering procedure that perfectly preserves minimum cuts. Recently, Li [Li21] gave an $m^{1+o(1)}$ deterministic minimum-cut algorithm for weighted graphs; this form of running time has been termed "almost-linear''. Li uses almost-linear time deterministic expander decompositions which do not perfectly preserve minimum cuts, but he can use these clusterings to, in a sense, "derandomize'' the methods of Karger. In terms of techniques, we provide a structural theorem that says there exists a sparse clustering that preserves minimum cuts in a weighted graph with $o(1)$ error. In addition, we construct it deterministically in near linear time. This was done exactly for simple graphs in [KT19, HRW20] and with polylogarithmic error for weighted graphs in [Li21]. Extending the techniques in [KT19, HRW20] to weighted graphs presents significant challenges, and moreover, the algorithm can only polylogarithmically approximately preserve minimum cuts. A remaining challenge is to reduce the polylogarithmic-approximate clusterings to $1+o(1/\log n)$-approximate so that they can be applied recursively as in [Li21] over $O(\log n)$ many levels. This is an additional challenge that requires building on properties of tree-packings in the presence of a wide range of edge weights to, for example, find sources for local flow computations which identify minimum cuts that cross clusters.

Figures (2)

• Figure 1: The set $U^\dag$ is uncrossed to $U^*\subseteq U^\dag\cap C_i$ according to property (\ref{['item:isolating-cuts-3']}) of \ref{['lem:isolating-cuts']}. The blue edges have weight at most $(1+\epsilon)$ times the black edges.
• Figure 2: The variables defined for the remainder of the proof. The black ellipse represents the cluster $A$ and the bottom half (below the black diagonal) is the set $U$.

Theorems & Definitions (92)

• Definition 2.1: Strength
• Definition 2.3
• Lemma 3.0: $s_0$-strong partition
• Lemma 3.0: Large cluster decomposition
• Lemma 3.0: Small cluster decomposition
• Lemma 3.1: Progress per iteration
• proof
• Lemma 3.2: Structure lemma
• Claim 3.3
• proof