Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Faster Pseudo-Deterministic Minimum Cut

Yotam Kenneth-Mordoch

TL;DR

A natural graph-theoretic tie-breaking mechanism that uniquely selects a canonical minimum cut in a fully-dynamic unweighted graph and improved pseudo-deterministic algorithms for unweighted graphs in the dynamic streaming and cut-query models of computation, matching the best randomized algorithms.

Abstract

Pseudo-deterministic algorithms are randomized algorithms that, with high constant probability, output a fixed canonical solution. The study of pseudo-deterministic algorithms for the global minimum cut problem was recently initiated by Agarwala and Varma [ITCS'26], who gave a black-box reduction incurring an $O(\log n \log \log n)$ overhead. We introduce a natural graph-theoretic tie-breaking mechanism that uniquely selects a canonical minimum cut. Using this mechanism, we obtain: (i) A pseudo-deterministic minimum cut algorithm for weighted graphs running in $O(m\log^2 n)$ time, eliminating the $O(\log n \log \log n)$ overhead of prior work and matching existing randomized algorithms. (ii) The first pseudo-deterministic algorithm for maintaining a canonical minimum cut in a fully-dynamic unweighted graph, with $\mathrm{polylog}(n)$ update time and $\tilde{O}(n)$ query time. (iii) Improved pseudo-deterministic algorithms for unweighted graphs in the dynamic streaming and cut-query models of computation, matching the best randomized algorithms.

Faster Pseudo-Deterministic Minimum Cut

TL;DR

A natural graph-theoretic tie-breaking mechanism that uniquely selects a canonical minimum cut in a fully-dynamic unweighted graph and improved pseudo-deterministic algorithms for unweighted graphs in the dynamic streaming and cut-query models of computation, matching the best randomized algorithms.

Abstract

Pseudo-deterministic algorithms are randomized algorithms that, with high constant probability, output a fixed canonical solution. The study of pseudo-deterministic algorithms for the global minimum cut problem was recently initiated by Agarwala and Varma [ITCS'26], who gave a black-box reduction incurring an overhead. We introduce a natural graph-theoretic tie-breaking mechanism that uniquely selects a canonical minimum cut. Using this mechanism, we obtain: (i) A pseudo-deterministic minimum cut algorithm for weighted graphs running in time, eliminating the overhead of prior work and matching existing randomized algorithms. (ii) The first pseudo-deterministic algorithm for maintaining a canonical minimum cut in a fully-dynamic unweighted graph, with update time and query time. (iii) Improved pseudo-deterministic algorithms for unweighted graphs in the dynamic streaming and cut-query models of computation, matching the best randomized algorithms.
Paper Structure (19 sections, 23 theorems, 2 equations, 1 table, 1 algorithm)

This paper contains 19 sections, 23 theorems, 2 equations, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Results
  3. Technical Overview
  4. Pseudo-Deterministic Minimum Cut
  5. Pseudo-deterministic fully-dynamic algorithm.
  6. Additional models of computation.
  7. Preliminaries
  8. Subtree Minimum Query
  9. Link-Cut Tree
  10. Pseudo-Deterministic Minimum-Cut Algorithm
  11. $1$-Respecting Cut Case
  12. Descendant Case
  13. Case 1: all minimum descendant cuts exclude $LN(u^{\downarrow})$.
  14. Case 2: there exists a minimum descendant cut that includes $LN(u^{\downarrow})$.
  15. Independent Case
  16. ...and 4 more sections

Key Result

Theorem 1.1

There exists a randomized algorithm that returns a canonical minimum cut in a weighted graph $G$ in $O(m\log^2 n)$ time with high probability.

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Claim 1.5
  • proof
  • Corollary 1.6: HHS24
  • Lemma 1.7
  • Definition 1.8: Non-Trivial Minimum Cut Sparsifier (NMC)
  • Theorem 1.9: Theorem 1 of HKMR25
  • ...and 37 more