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 is an element of V (G) and an integer k, one can in randomized k(O(1)) center dot (|V (G)| + |E(G)|) time sample a set A subset of ((V (G))(2)) such that the following holds: for every inclusionwise 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 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.