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^𝒪(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^{-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(k4 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.

我们提出了最近引入的流量增强技术的无向版本：给定一个无向多重图G ，其具有可区分的顶点s,t ∈ V(G)和一个整数k，我们可以随机化k ^𝒪(1) ⋅ (|V(G) )| + |E(G)|)对集合A ⊆ \(\binom{V(G)}{2}\) 进行时间采样，使得以下内容成立：对于G中的每个包含明智的最小st割Z基数至多为k，Z成为G+A中s和t之间的最小基数割（即，在添加了A的所有边的多重图G中），概率为 2 ^{-𝒪( k log k} )。

与有向图 [STOC 2022] 的版本相比，此处提供的版本提高了成功概率（2 ^{-𝒪( k log k} ) 而不是 2 ^{-𝒪( k 4 log k )}），线性依赖于图大小运行时间限制，以及可以说是更简单的证明。

直接的推论是，双目标st -Cut问题可以在无向图上以随机 FPT 时间 2 ^{𝒪( k log k )} (|V(G)|+|E(G)|)解决。