Deterministic and Exact Fully-dynamic Minimum Cut of Superpolylogarithmic Size in Subpolynomial Time

TL;DR

This paper introduces a deterministic fully-dynamic minimum cut algorithm with subpolynomial update time for small cut sizes, and extends it to approximate weighted graphs, advancing dynamic graph algorithms.

Contribution

It presents the first deterministic fully-dynamic minimum cut algorithm for certain small cut sizes and combines it with graph sparsification for approximate solutions.

Findings

Deterministic algorithm with $n^{o(1)}$ update time for small minimum cuts.

Extension to $(1+)$-approximate algorithms on weighted graphs.

Replaces randomized procedures with deterministic local minimum cut techniques.

Abstract

We present an exact fully-dynamic minimum cut algorithm that runs in $n^{o(1)}$ deterministic update time when the minimum cut size is at most $2^{\Theta(\log^{3/4-c}n)}$ for any $c>0$, improving on the previous algorithm of Jin, Sun, and Thorup (SODA 2024) whose minimum cut size limit is $(\log n)^{o(1)}$. Combined with graph sparsification, we obtain the first $(1+\epsilon)$-approximate fully-dynamic minimum cut algorithm on weighted graphs, for any $\epsilon\ge2^{-\Theta(\log^{3/4-c}n)}$, in $n^{o(1)}$ randomized update time. Our main technical contribution is a deterministic local minimum cut algorithm, which replaces the randomized LocalKCut procedure from El-Hayek, Henzinger, and Li (SODA 2025).