We introduce a variant of the multiway cut that we call the min-max connected multiway cut. Given a graph $G=(V,E)$ and a set $Γ\subseteq V$ of $t$ terminals, partition $V$ into $t$ parts such that each part is connected and contains exactly one terminal; the objective is to minimize the maximum weight of the edges leaving any part of the partition. This problem is a natural modification of the standard multiway cut problem and it differs from it in two ways: first, the cost of a partition is defined to be the maximum size of the boundary of any part, as opposed to the sum of all boundaries, and second, the subgraph induced by each part is required to be connected. Although the modified objective function has been considered before in the literature under the name min-max multiway cut, the requirement on each component to be connected has not been studied as far as we know. We show various hardness results for this problem, including a proof of weak NP-hardness of the weighted version of the problem on graphs with tree-width two, and provide a pseudopolynomial time algorithm as well as an FPTAS for the weighted problem on trees. As a consequence of our investigation we also show that the (unconstrained) min-max multiway cut problem is NP-hard even for three terminals, strengthening the known results.

翻译：我们引入一种多路割的变体，称为最小最大连通多路割。给定图$G=(V,E)$与包含$t$个终端的集合$Γ\subseteq V$，将$V$划分为$t$个子集，使得每个子集连通且恰好包含一个终端；目标是最小化划分中任意子集导出边界的最大权重。该问题是标准多路割问题的自然修正，其差异主要体现在两方面：首先，划分的成本定义为任意子集边界的最大规模（而非所有边界之和）；其次，要求每个子集导出的子图必须连通。虽然修正后的目标函数在已有文献中以最小最大多路割的名称被研究过，但据我们所知，各分量需连通的要求尚未被探讨。我们证明了该问题的若干困难性结果，包括树宽为二的图上加权版本具有弱NP难度的证明，并针对树上的加权问题提出了伪多项式时间算法与完全多项式时间近似方案。通过本研究，我们还进一步证明了（无约束的）最小最大多路割问题即使对于三个终端也是NP难的，从而强化了现有结论。