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. Our main motivation for studying this problems is its close connection with the spanning tree congestion problem that has been extensively studied but on which little progress has been made. 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 treewidth two, and $W[1]$-hardness for the problem when parameterized by the treewidth of the graph. Complementing our lower bounds tightly, we also provide a pseudopolynomial time algorithm as well as an FPTAS for the weighted problem on graphs of bounded treewidth. 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：我们提出了一种多路割的变体，称为最小-最大连通多路割。给定图 $G=(V,E)$ 和包含 $t$ 个终端的集合 $Γ\subseteq V$，将 $V$ 划分为 $t$ 个部分，使得每个部分连通且恰好包含一个终端；目标是最小化离开任一剖分部分的边的最大权重。该问题是标准多路割问题的自然修正，与之存在两点不同：首先，剖分的代价定义为所有部分边界的最大规模，而非所有边界之和；其次，要求每个部分诱导的子图是连通的。尽管此前文献中已以“最小-最大多路割”之名考虑过修改后的目标函数，但据我们所知，关于每个分量必须连通的要求尚未被研究。我们研究此问题的主要动机在于其与生成树拥塞问题的密切联系——该问题虽已被广泛研究，但进展甚微。我们对这一问题展示了多种困难性结果，包括在树宽为二的图上加权版本是弱NP难的证明，以及当以图树宽为参数时问题属于 $W[1]$-困难。为了紧致地补充下界，我们还为有界树宽图上的加权问题提供了伪多项式时间算法及完全多项式时间近似方案（FPTAS）。作为研究的推论，我们还证明了（无约束的）最小-最大多路割问题即使对三个终端也是NP难的，这强化了已知结果。