Faster Pseudo-Deterministic Minimum Cut

TL;DR

This paper introduces a natural tie-breaking mechanism for pseudo-deterministic algorithms that efficiently finds and maintains a canonical minimum cut in various graph models, matching randomized algorithms' performance.

Contribution

It presents the first natural tie-breaking method for pseudo-deterministic minimum cut algorithms, enabling improved algorithms for weighted, dynamic, and streaming graph models.

Findings

Achieved $O(m ext{log}^2 n)$ time for weighted graphs, matching randomized algorithms.

Developed the first pseudo-deterministic fully-dynamic minimum cut algorithm with polylog update time.

Matched the best randomized algorithms in dynamic streaming and cut-query models.

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.