Flow-augmentation II: Undirected graphs

Abstract

We present an undirected version of the recently introduced flow-augmentation technique: Given an undirected multigraph G with distinguished vertices s , t ∈ V ( G ) and an integer k , one can in randomized \(k^{\mathcal {O}(1)} \cdot (|V(G)| + |E(G)|) \) 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 \(2^{-\mathcal {O}(k \log k)} \) . Compared to the version for directed graphs [STOC 2022], the version presented here has improved success probability ( \(2^{-\mathcal {O}(k \log k)} \) instead of \(2^{-\mathcal {O}(k^4 \log k)} \) ), 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 \(2^{\mathcal {O}(k \log k)} (|V(G)|+|E(G)|) \) on undirected graphs.