The Multicut problem asks for a minimum cut separating certain pairs of vertices: formally, given a graph $G$ and demand graph $H$ on a set $T\subseteq V(G)$ of terminals, the task is to find a minimum-weight set $C$ of edges of $G$ such that whenever two vertices of $T$ are adjacent in $H$, they are in different components of $G\setminus C$. Colin de Verdi\`{e}re [Algorithmica, 2017] showed that Multicut with $t$ terminals on a graph $G$ of genus $g$ can be solved in time $f(t,g)n^{O(\sqrt{g^2+gt+t})}$. Cohen-Addad et al. [JACM, 2021] proved a matching lower bound showing that the exponent of $n$ is essentially best possible (for fixed values of $t$ and $g$), even in the special case of Multiway Cut, where the demand graph $H$ is a complete graph. However, this lower bound tells us nothing about other special cases of Multicut such as Group 3-Terminal Cut. We show that if the demand pattern is, in some sense, close to being a complete bipartite graph, then Multicut can be solved faster than $f(t,g)n^{O(\sqrt{g^2+gt+t})}$, and furthermore this is the only property that allows such an improvement. Formally, for a class $\mathcal{H}$ of graphs, Multicut$(\mathcal{H})$ is the special case where the demand graph $H$ is in $\mathcal{H}$. For every fixed class $\mathcal{H}$ (satisfying some mild closure property), fixed $g$, and fixed $t$, our main result gives tight upper and lower bounds on the exponent of $n$ in algorithms solving Multicut$(\mathcal{H})$. In addition, we investigate a similar setting where, instead of parameterizing by the genus $g$ of $G$, we parameterize by the minimum number $k$ of edges of $G$ that need to be deleted to obtain a planar graph. Interestingly, in this setting it makes a significant difference whether the graph $G$ is weighted or unweighted: further nontrivial algorithmic techniques give substantial improvements in the unweighted case.

翻译：多割问题要求找到一个最小割集，用于分离某些指定的顶点对：形式化地，给定图 $G$ 和终端集 $T\subseteq V(G)$ 上的需求图 $H$，任务是找到 $G$ 中边的最小权重集 $C$，使得每当 $T$ 中两个顶点在 $H$ 中相邻时，它们在 $G\setminus C$ 中属于不同连通分量。Colin de Verdière [Algorithmica, 2017] 表明，在亏格为 $g$ 的图 $G$ 上，具有 $t$ 个端点的多割问题可在时间 $f(t,g)n^{O(\sqrt{g^2+gt+t})}$ 内求解。Cohen-Addad 等人 [JACM, 2021] 证明了匹配的下界，表明 $n$ 的指数本质上是可能的最优值（对于固定的 $t$ 和 $g$ 值），即使在多路割（即需求图 $H$ 为完全图）的特殊情况下也是如此。然而，此下界并未揭示多割问题的其他特殊情况（如分组三终端割）的本质。我们表明，如果需求模式在某种意义上接近完全二分图，那么多割问题可以比 $f(t,g)n^{O(\sqrt{g^2+gt+t})}$ 更快地求解，并且这一性质是允许此类改进的唯一条件。形式上，对于图类 $\mathcal{H}$，多割$(\mathcal{H})$ 是需求图 $H$ 属于 $\mathcal{H}$ 的特殊情况。对于每个固定的图类 $\mathcal{H}$（满足某些温和闭包性质）、固定的 $g$ 和固定的 $t$，我们的主要结果给出了求解多割$(\mathcal{H})$ 算法中 $n$ 的指数的严格上、下界。此外，我们研究了类似设定，其中参数化方式由 $G$ 的亏格 $g$ 切换为将 $G$ 变为平面图所需删除的最小边数 $k$。有趣的是，在此设定中，图 $G$ 是否加权会产生显著差异：非平凡算法技术在无权重情况下带来了实质性的改进。