Flow-augmentation II: Undirected graphs

TL;DR

This paper introduces an improved undirected flow-augmentation technique that efficiently samples edge sets to identify minimal $st$-cuts, with higher success probability and simpler proof, enabling faster algorithms for related problems.

Contribution

It presents an undirected version of flow-augmentation with better success probability, linear time dependency, and a simpler proof, extending prior directed graph results.

Findings

Success probability improved to 2^{-O(k log k)}

Linear dependency on graph size in running time

Enables randomized FPT algorithms for Bi-objective $st$-Cut

Abstract

We present an undirected version of the recently introduced flow-augmentation technique: Given an undirected multigraph $G$ with distinguished vertices $s,t \in V(G)$ and an integer $k$, one can in randomized $k^{O(1)} \cdot (|V(G)| + |E(G)|)$ time sample a set $A \subseteq \binom{V(G)}{2}$ such that the following holds: for every inclusion-wise minimal $st$-cut $Z$ in $G$ of cardinality at most $k$, $Z$ becomes a minimum-cardinality cut between $s$ and $t$ in $G+A$ (i.e., in the multigraph $G$ with all edges of $A$ added) with probability $2^{-O(k \log k)}$. Compared to the version for directed graphs [STOC 2022], the version presented here has improved success probability ($2^{-O(k \log k)}$ instead of $2^{-O(k^4 \log k)}$), linear dependency on the graph size in the running time bound, and an arguably simpler proof. An immediate corollary is that the Bi-objective $st$-Cut problem can be solved in randomized FPT time $2^{O(k \log k)} (|V(G)|+|E(G)|)$ on undirected graphs.