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

2025-12-15

AI Summary

This paper addresses the fully dynamic minimum $k$-cut problem for super-logarithmic $k$ in general graphs, presenting the first deterministic fully dynamic algorithm. Methodologically, it introduces: (1) a deterministic local $k$-cut algorithm, overcoming the prior reliance on randomization in LocalKCut; and (2) a synthesis of graph sparsification, hierarchical contraction structures, and subpolynomial-time analysis techniques. Key contributions include: raising the largest $k$ for which exact dynamic minimum $k$-cut is supported—from $(log n)^{o(1)}$ to $2^{Theta(log^{3/4-c} n)}$—while achieving $n^{o(1)}$ deterministic update time; and, integrating sparsification, obtaining the first fully dynamic $(1+varepsilon)$-approximation algorithm with $n^{o(1)}$ randomized update time, where $varepsilon geq 2^{-Theta(log^{3/4-c} n)}$.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 $2^{Θ(log^{3/4-c}n)}$ 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 $(1+ε)$-approximate fully-dynamic minimum cut algorithm on weighted graphs, for any $εge2^{-Θ(log^{3/4-c}n)}$, 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).

Problem

• Exact fully-dynamic minimum cut algorithm
• Deterministic subpolynomial update time
• Approximate minimum cut for weighted graphs

Innovation

• Deterministic local minimum cut algorithm
• Exact fully-dynamic algorithm with subpolynomial time
• First approximate fully-dynamic algorithm for weighted graphs