Fully Dynamic s-t Edge Connectivity in Subpolynomial Time

TL;DR

This paper introduces a deterministic fully dynamic algorithm capable of answering $c$-edge connectivity queries with subpolynomial worst-case update and query times for graphs with $n$ vertices, extending previous polylogarithmic results.

Contribution

It develops a new multi-level sparsification framework and a novel update algorithm for $c$-edge connectivity, achieving subpolynomial update times in the dynamic setting.

Findings

Achieves $n^{o(1)}$ worst-case update and query time for $c$-edge connectivity.

Extends the $c$-edge connectivity vertex sparsifier to a multi-level framework.

Provides a new update algorithm with subpolynomial time complexity.

Abstract

We present a deterministic fully dynamic algorithm to answer $c$-edge connectivity queries on pairs of vertices in $n^{o(1)}$ worst case update and query time for any positive integer $c = (\log n)^{o(1)}$ for a graph with $n$ vertices. Previously, only polylogarithmic and $O(\sqrt{n})$ worst case update time fully dynamic algorithms were known for answering $1$, $2$ and $3$-edge connectivity queries respectively [Henzinger and King 1995, Frederikson 1997, Galil and Italiano 1991]. Our result extends the $c$-edge connectivity vertex sparsifier [Chalermsook et al. 2021] to a multi-level sparsification framework. As our main technical contribution, we present a novel update algorithm for the multi-level $c$-edge connectivity vertex sparsifier with subpolynomial update time.