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

Yotam Kenneth-Mordoch, Robert Krauthgamer

TL;DR

The paper solves all-pairs minimum cut in the cut-query model by constructing a Gomory–Hu tree with $\tilde{O}(n^{7/4})$ cut queries. It introduces weak isolating cuts and a novel $(\tau,F)$-star contraction to handle unfriendly cuts, and develops a parallel, subproblem–driven single-source min-cut routine that feeds the Gomory–Hu refinement. Central to the approach are two data structures: a friendly cut sparsifier via expander decomposition and a contraction-based reduction that preserves critical minimum cuts. The results close part of the gap between all-pairs and single-pair cut queries, advancing submodular function minimization in the cut-query model and suggesting avenues for weighted graphs and related models. Overall, the work provides a near-optimal subquadratic query framework for recovering the Gomory–Hu representation and hence the entire minimum-cut landscape of the graph.

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 $\tilde{O}(n^{7/4})$ cut queries.

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

TL;DR

The paper solves all-pairs minimum cut in the cut-query model by constructing a Gomory–Hu tree with cut queries. It introduces weak isolating cuts and a novel -star contraction to handle unfriendly cuts, and develops a parallel, subproblem–driven single-source min-cut routine that feeds the Gomory–Hu refinement. Central to the approach are two data structures: a friendly cut sparsifier via expander decomposition and a contraction-based reduction that preserves critical minimum cuts. The results close part of the gap between all-pairs and single-pair cut queries, advancing submodular function minimization in the cut-query model and suggesting avenues for weighted graphs and related models. Overall, the work provides a near-optimal subquadratic query framework for recovering the Gomory–Hu representation and hence the entire minimum-cut landscape of the graph.

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 with vertices, our randomized algorithm constructs a Gomory-Hu tree of , and thus solves the all-pairs minimum cut problem, using cut queries.
Paper Structure (21 sections, 22 theorems, 22 equations, 3 algorithms)

This paper contains 21 sections, 22 theorems, 22 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Technical Overview
  3. Related Work
  4. Future Work
  5. Preliminaries
  6. Cut Query Primitives
  7. Edge Perturbation
  8. Cut Sparsification
  9. Isolating Cuts
  10. (tau,F)-Star Contraction
  11. Proof of Claim \ref{['claim:expectation-c-r-ratio']}
  12. Single-Source Minimum Cuts
  13. Proof of Claim \ref{['claim:unfriendly-min-cut']}
  14. Friendly Cut Sparsifiers
  15. Expander Decomposition
  16. ...and 6 more sections

Key Result

Theorem 1.1

There exists a randomized algorithm for constructing a Gomory-Hu tree of an unweighted graph $G=(V,E)$ with $n$ vertices, that uses $\tilde{O}(n^{7/4})$ cut queries and succeeds with probability $1-1/\mathop{\mathrm{poly}}\nolimits(n)$.

Theorems & Definitions (54)

  • Theorem 1.1
  • Lemma 1.1
  • Lemma 1.1: Single-Source Minimum Cut
  • Definition 1.2
  • Lemma 1.3: Weak Isolating Cuts
  • Definition 1.4
  • Lemma 1.5: Friendly Cut Sparsifier
  • Definition 1.6
  • Proposition 1.7: Proposition 4.1 in AEGLMN22
  • Lemma 1.8: Guarantees for $(\tau,F)$-star contraction
  • ...and 44 more