AI 中文总结
本文证明以$k$为参数的多路$f$-割问题是固定参数可解的，该结论涵盖图上边多路割的经典结果，且其正确性证明是该事实已知的最简单初等证明。
AI 中文摘要
有限集合$E$上的连通性函数$f$是满足次模、对称且$f(\boldsymbol{\theta})=0$的函数，形式为$f\boldsymbol{\theta}:2^E\to\boldsymbol{Z}$。给定通过值谕示给出的连通性函数$f$、终端$t_1,\boldsymbol{\theta},t_r\boldsymbol{\theta}\boldsymbol{\theta}\boldsymbol{\theta}E$及整数$k$，多路$f$-割问题要求判断是否存在$E$的划分$(P_1,\boldsymbol{\theta},P_r)$，使得对每个$i$有$t_i\boldsymbol{\theta}P_i$且$\boldsymbol{\theta}_{i=1}^r f(P_i)\boldsymbol{\theta}k$。本文证明，以$k$为参数的多路$f$-割问题是固定参数可解的。图的割函数属于连通性函数，因此作为特例，本文得到图上的经典边多路割问题是固定参数可解的结果。本文的正确性证明完全初等，可说是该事实已知的最简单证明。
英文摘要
A connectivity function on a finite set $E$ is a function $f\colon 2^E\to\mathbb Z$ that is submodular and symmetric, with $f(\varnothing)=0$. Given a connectivity function $f$ via a value oracle, terminals $t_1,\ldots,t_r\in E$, and an integer $k$, the Multiway $f$-Cut problem asks whether $E$ has a partition $(P_1,\ldots,P_r)$ with $t_i\in P_i$ for every $i$ and $\sum_{i=1}^r f(P_i)\le k$. We prove that Multiway $f$-Cut is fixed-parameter tractable parameterized by $k$. Cut functions of graphs are connectivity functions, so as a special case we recover the classical result that Edge Multiway Cut in graphs is fixed-parameter tractable. Our proof of correctness is completely elementary, and is arguably the simplest known proof of this fact.
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