Faster All-Pairs Minimum Cut: Bypassing Exact Max-Flow

TL;DR

This work introduces an all-pairs minimum cut sparsifier that preserves all minimum $s,t$-cuts while adding proxy vertices, enabling faster algorithms across cut-query, streaming, and fully-dynamic models by bypassing exact max-flow. The core construction combines a friendly cut sparsifier, built via expander decomposition and contraction-based refinements, with a star transform that returns an APMC sparsifier $H_{ ext{ap}}$ whose edges satisfy $|E_{ ext{ap}}| leq |E_{ ext{fr}}|+n$. The approach yields a randomized $ ilde{O}(n^{3/2})$-query cut-query algorithm, a deterministic $n^{3/2+o(1)}$-time fully-dynamic algorithm maintaining a Gomory-Hu tree, and a two-pass dynamic-streaming method using $ ilde{O}(n^{3/2})$ space, all improving previous bounds in their respective models. The framework is robust to edge updates and extends to dynamic streams via power-cut sparsifiers, though extending to weighted graphs remains open. Overall, the paper provides a modular, update-robust route to exact APMC via approximate min-cuts, with practical implications for fast cut-based graph analysis and submodular minimization connections.

Abstract

All-Pairs Minimum Cut (APMC) is a fundamental graph problem that asks to find a minimum $s,t$-cut for every pair of vertices $s,t$. A recent line of work on fast algorithms for APMC has culminated with a reduction of APMC to $\mathrm{polylog}(n)$-many max-flow computations. But unfortunately, no fast algorithms are currently known for exact max-flow in several standard models of computation, such as the cut-query model and the fully-dynamic model. Our main technical contribution is a sparsifier that preserves all minimum $s,t$-cuts in an unweighted graph, and can be constructed using only approximate max-flow computations. We then use this sparsifier to devise new algorithms for APMC in unweighted graphs in several computational models: (i) a randomized algorithm that makes $\tilde{O}(n^{3/2})$ cut queries to the input graph; (ii) a deterministic fully-dynamic algorithm with $n^{3/2+o(1)}$ worst-case update time; and (iii) a randomized two-pass streaming algorithm with space requirement $\tilde{O}(n^{3/2})$. These results improve over the known bounds, even for (single pair) minimum $s,t$-cut in the respective models.

Figures (1)

• Figure 1: An illustration of the star-transform operation. The super-vertex $u={\{v_1,\ldots,v_5\}}$ is split into its constituent vertices, adding a proxy vertex $u'$ that is connected to every vertex $v_i\in u$ with edge weight equal to its degree in the induced graph $G[u]$. For example, $v_5$ has two neighbors in $G[u]$, hence it is connected to $u'$ with an edge of weight $2$.

Theorems & Definitions (54)

• Definition 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4: Query Complexity Version of GKYY25
• Theorem 1.5
• Corollary 1.6
• Theorem 1.7
• Lemma 1.8
• Theorem 2.1
• Remark 2.2