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

2025-11-13

AI Summary

All-Pairs Minimum Cut (APMC) is a fundamental graph problem; classical approaches rely on repeatedly computing exact maximum flows, yet no efficient algorithms exist for the cut-query, fully dynamic, and streaming models. This paper introduces the first APMC framework for unweighted graphs that bypasses exact maximum flow computations. Our approach constructs a *minimum-cut sparsifier*—a sparse graph preserving all minimum cuts—and computes APMC using only approximate maximum flows. Integrating graph sparsification, randomization, and dynamic maintenance techniques, we achieve state-of-the-art results across multiple models: $ ilde{O}(n^{3/2})$ cut queries in the cut-query model; worst-case $n^{3/2+o(1)}$ update time in the fully dynamic model; and a two-pass streaming algorithm with $ ilde{O}(n^{3/2})$ space. To our knowledge, this is the first efficient APMC solution that avoids exact maximum flow computation entirely.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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 $ ilde{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 $ ilde{O}(n^{3/2})$. These results improve over the known bounds, even for (single pair) minimum $s,t$-cut in the respective models.

Problem

• Bypassing exact max-flow computations for All-Pairs Minimum Cut
• Developing efficient algorithms for APMC in cut-query and dynamic models
• Constructing sparsifiers preserving minimum cuts using approximate max-flow

Innovation

• Sparsifier preserves all minimum cuts
• Constructed using approximate max-flow computations
• Applied in cut-query, dynamic, and streaming models