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.

翻译：我们提出了近期引入的流增强技术的无向版本：给定一个无向多重图 $G$，其中包含指定顶点 $s,t \in V(G)$ 和一个整数 $k$，可以在随机化 $k^{O(1)} \cdot (|V(G)| + |E(G)|)$ 时间内采样一个集合 $A \subseteq \binom{V(G)}{2}$，使得以下性质成立：对于 $G$ 中每个包含关系意义下最小且基数至多为 $k$ 的 $st$-割 $Z$，$Z$ 在 $G+A$（即添加了 $A$ 中所有边的多重图 $G$）中成为 $s$ 和 $t$ 之间的最小基数割的概率为 $2^{-O(k \log k)}$。与有向图版本 [STOC 2022] 相比，本文提出的版本具有更高的成功概率（$2^{-O(k \log k)}$ 而非 $2^{-O(k^4 \log k)}$）、运行时间界限中关于图大小的线性依赖性，以及一个更简洁的证明。一个直接推论是，双目标 $st$-割问题可以在无向图上以随机化 FPT 时间 $2^{O(k \log k)} (|V(G)|+|E(G)|)$ 解决。