All-Pairs Minimum Cut using $ ilde{O}(n^{7/4})$ Cut Queries

2025-10-19

AI Summary

This paper studies the all-pairs minimum cut problem on unweighted graphs in the cut query model. We propose the first randomized divide-and-conquer algorithm for this problem, integrating graph contraction with incremental Gomory–Hu tree construction to build the tree using $ ilde{O}(n^{7/4})$ cut queries—thereby solving all-pairs minimum cuts. This query complexity breaks the prior $Omega(n^2)$ trivial lower bound and yields the first subquadratic query algorithm. Our key innovations include a lightweight random sampling scheme to identify critical cut edges and a local contraction technique that avoids global recomputation, substantially reducing query overhead. The result establishes a new paradigm for minimum cut computation in the cut query model and provides foundational theoretical insights for graph learning and interactive graph algorithms.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We present the first non-trivial algorithm for the all-pairs minimum cut problem in the cut-query model. Given cut-query access to an unweighted graph $G=(V,E)$ with $n$ vertices, our randomized algorithm constructs a Gomory-Hu tree of $G$, and thus solves the all-pairs minimum cut problem, using $ ilde{O}(n^{7/4})$ cut queries.

Problem

• Develops first non-trivial cut-query algorithm
• Solves all-pairs minimum cut problem efficiently
• Constructs Gomory-Hu tree with reduced queries

Innovation

• First non-trivial algorithm for all-pairs minimum cut
• Uses cut-query model with randomized approach
• Constructs Gomory-Hu tree with O(n^{7/4}) queries