Flow-augmentation II: Undirected Graphs

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 ^????(1) ⋅ (|V(G)| + |E(G)|) time sample a set A ⊆ \(\binom{V(G)}{2}\) 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 ^{-????( k log k} ). Compared to the version for directed graphs [STOC 2022], the version presented here has improved success probability (2 ^{-????( 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 ^{????( k log k )} (|V(G)|+|E(G)|) on undirected graphs.