We consider variants of the classic Multiway Cut problem. Multiway Cut asks to partition a graph $G$ into $k$ parts so as to separate $k$ given terminals. Recently, Chandrasekaran and Wang (ESA 2021) introduced $\ell_p$-norm Multiway, a generalization of the problem, in which the goal is to minimize the $\ell_p$ norm of the edge boundaries of $k$ parts. We provide an $O(\log^{1/2} n\log^{1/2+1/p} k)$ approximation algorithm for this problem, improving upon the approximation guarantee of $O(\log^{3/2} n \log^{1/2} k)$ due to Chandrasekaran and Wang. We also introduce and study Norm Multiway Cut, a further generalization of Multiway Cut. We assume that we are given access to an oracle, which answers certain queries about the norm. We present an $O(\log^{1/2} n \log^{7/2} k)$ approximation algorithm with a weaker oracle and an $O(\log^{1/2} n \log^{5/2} k)$ approximation algorithm with a stronger oracle. Additionally, we show that without any oracle access, there is no $n^{1/4-\varepsilon}$ approximation algorithm for every $\varepsilon > 0$ assuming the Hypergraph Dense-vs-Random Conjecture.

翻译：我们考虑经典多路割问题的变体。多路割问题要求将图$G$划分为$k$个部分，以分离$k$个给定终端。最近，Chandrasekaran和Wang (ESA 2021) 引入了该问题的推广形式——$\ell_p$范数多路割，其目标是最小化$k$个部分的边界的$\ell_p$范数。我们为该问题提供了一个$O(\log^{1/2} n\log^{1/2+1/p} k)$近似算法，改进了Chandrasekaran和Wang得到的$O(\log^{3/2} n \log^{1/2} k)$近似保证。我们还引入并研究了范数多路割，这是多路割问题的进一步推广。我们假设可以访问一个预言机，该预言机能够回答关于范数的某些查询。我们提出了一个使用较弱预言机的$O(\log^{1/2} n \log^{7/2} k)$近似算法和一个使用较强预言机的$O(\log^{1/2} n \log^{5/2} k)$近似算法。此外，我们证明了在假设超图稠密-随机猜想成立的情况下，对于任意$\varepsilon > 0$，若无预言机访问，则不存在$n^{1/4-\varepsilon}$近似算法。