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Fully Dynamic Min-Cut of Superconstant Size in Subpolynomial Time

Wenyu Jin, Xiaorui Sun, Mikkel Thorup

TL;DR

The paper addresses the fully dynamic minimum $c$-cut problem on undirected graphs for $c = (\log n)^{o(1)}$, aiming to output a global minimum cut of size at most $c$ after each online update. It introduces a localization framework that uses expander decompositions and terminal $c$-edge connectivity sparsifiers to reduce global minimization to tractable subproblems, and then combines a terminal-cut data structure with a carefully maintained set of local cuts via a multi-level sparsifier. The main contributions are a deterministic fully dynamic algorithm with subpolynomial update time $n^{o(1)}$ and rigorous mechanisms to recover a global minimum cut when it exists, along with a decremental corollary that yields efficient maintenance of a minimum $c$-edge connectivity structure. This approach advances the state of dynamic graph algorithms by achieving subpolynomial-time updates for exact minimum-cut computation in the regime of small cuts, with potential implications for dynamic network reliability and related optimization tasks.

Abstract

We present a deterministic fully dynamic algorithm with subpolynomial worst-case time per graph update such that after processing each update of the graph, the algorithm outputs a minimum cut of the graph if the graph has a cut of size at most $c$ for some $c = (\log n)^{o(1)}$. Previously, the best update time was $\widetilde O(\sqrt{n})$ for any $c > 2$ and $c = O(\log n)$ [Thorup, Combinatorica'07].

Fully Dynamic Min-Cut of Superconstant Size in Subpolynomial Time

TL;DR

The paper addresses the fully dynamic minimum -cut problem on undirected graphs for , aiming to output a global minimum cut of size at most after each online update. It introduces a localization framework that uses expander decompositions and terminal -edge connectivity sparsifiers to reduce global minimization to tractable subproblems, and then combines a terminal-cut data structure with a carefully maintained set of local cuts via a multi-level sparsifier. The main contributions are a deterministic fully dynamic algorithm with subpolynomial update time and rigorous mechanisms to recover a global minimum cut when it exists, along with a decremental corollary that yields efficient maintenance of a minimum -edge connectivity structure. This approach advances the state of dynamic graph algorithms by achieving subpolynomial-time updates for exact minimum-cut computation in the regime of small cuts, with potential implications for dynamic network reliability and related optimization tasks.

Abstract

We present a deterministic fully dynamic algorithm with subpolynomial worst-case time per graph update such that after processing each update of the graph, the algorithm outputs a minimum cut of the graph if the graph has a cut of size at most for some . Previously, the best update time was for any and [Thorup, Combinatorica'07].
Paper Structure (25 sections, 24 theorems, 30 equations, 6 algorithms)

This paper contains 25 sections, 24 theorems, 30 equations, 6 algorithms.

Table of Contents

  1. Introduction
  2. Techniques
  3. A Static c-Edge Connectivity Algorithm via Localization
  4. Retrieving A Minimum c-Cut From Terminal Sparsifier
  5. Minimum c-Cut Data Structure
  6. Fully Dynamic Minimum c-Cut Algorithm
  7. Preliminaries
  8. Review: Multi-level Terminal Edge Connectivity Data Structure
  9. Terminal Edge Connectivity Sparsifier
  10. One-Level Terminal Edge Connectivity Data Structure
  11. Multi-Level Terminal Edge Connectivity Data Structure
  12. Minimum c-Cut Data Structure
  13. The Characterization of Minimum c-cuts
  14. Local Cut
  15. The Global Min-Cut Sparsifier Data Structure
  16. ...and 10 more sections

Key Result

Theorem 1.1

For any $c = (\log n)^{o(1)}$, there is a deterministic fully dynamic algorithm for an undirected graph with $n^{o(1)}$ running time per update such that after processing each update of the graph, the algorithm outputs a (global) minimum cut of the graph if the graph has a cut of size at most $c$, o

Theorems & Definitions (49)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1: Expander and Expander Decomposition
  • Theorem 2.2: Corollary 7.1 of chuzhoy2019deterministic
  • Theorem 2.3: Theorem 1.3 of saranurak2019expander, rephrased
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3: cut containment set
  • Lemma 3.4: js20
  • Lemma 3.5: js20
  • ...and 39 more