SIAM Journal on Computing, Volume 50, Issue 1, Page 1-31, January 2021.
For an undirected edge-weighted graph $G$ and a set $R$ of pairs of vertices called pairs of terminals, a multicut is a set of edges such that removing these edges from $G$ disconnects each pair in $R$. We provide an algorithm computing a $(1+\varepsilon)$-approximation of the minimum multicut of a graph $G$ in time $(g+t)^{(O(g+t)^3)}\cdot(1/\varepsilon)^{O(g+t)} \cdot n \log n$, where $g$ is the genus of $G$ and $t$ is the number of terminals. This is tight in several aspects, as the minimum multicut problem is both APX-hard and W[1]-hard (parameterized by the number of terminals), even on planar graphs (equivalently, when $g=0$). Our result, in the field of fixed-parameter approximation algorithms, mostly relies on concepts borrowed from computational topology of graphs on surfaces. In particular, we use and extend various recent techniques concerning homotopy, homology, and covering spaces. Interestingly, such topological techniques seem necessary even for the planar case. We also exploit classical ideas stemming from approximation schemes for planar graphs and low-dimensional geometric inputs. A key insight toward our result is a novel characterization of a minimum multicut as the union of some Steiner trees in the universal cover of the surface in which $G$ is embedded.

中文翻译：

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终端数量固定的嵌入式图多割的近线性近似方案

SIAM计算杂志，第50卷，第1期，第1-31页，2021年1月。
对于无向边加权图$ G $和称为顶点对的一组顶点对的一组$ R $，多边切割是一组边，这样从$ G $中删除这些边会使$ R $中的每对断开连接。我们提供了一种算法，可以在时间$（g + t）^ {（O（g + t）^ 3）} \ cdot {的情况下计算图$ G $的最小多重割的$（1+ \ varepsilon）$近似值。 1 / \ varepsilon）^ {O（g + t）} \ cdot n \ log n $，其中$ g $是$ G $的属，$ t $是终端的数目。这在几个方面都很严格，因为最小多点切割问题既是APX硬问题，也是W [1]硬问题（由终端数量参数化），即使在平面图上（等效地，当$ g = 0 $时）也是如此。在固定参数逼近算法领域，我们的结果主要依赖于从曲面图的计算拓扑中借用的概念。特别是，我们使用并扩展了有关同位性，同源性和覆盖空间的各种最新技术。有趣的是，即使对于平面情况，这种拓扑技术似乎也是必需的。我们还利用源自平面图和低维几何输入的近似方案的经典思想。对我们的结果的一个关键见解是最小多切面的新颖表征，它是嵌入$ G $的表面的通用覆盖物中的一些Steiner树的并集。