We study the Requirement Cut problem, a generalization of numerous classical graph partitioning problems including Multicut, Multiway Cut, $k$-Cut, and Steiner Multicut among others. Given a graph with edge costs, terminal groups $(S_1, ..., S_g)$ and integer requirements $(r_1,... , r_g)$; the goal is to compute a minimum-cost edge cut that separates each group $S_i$ into at least $r_i$ connected components. Despite many efforts, the best known approximation for Requirement Cut yields a double-logarithmic $O(\log(g).\log(n))$ approximation ratio as it relies on embedding general graphs into trees and solving the tree instance. In this paper, we explore two largely unstudied structural parameters in order to obtain single-logarithmic approximation ratios: (1) the number of minimal Steiner trees in the instance, which in particular is upper-bounded by the number of spanning trees of the graphs multiplied by $g$, and (2) the depth of series-parallel graphs. Specifically, we show that if the number of minimal Steiner trees is polynomial in $n$, then a simple LP-rounding algorithm yields an $O(\log n)$-approximation, and if the graph is series-parallel with a constant depth then a refined analysis of a known probabilistic embedding yields a $O(depth.\log(g))$-approximation on series-parallel graphs of bounded depth. Both results extend the known class of graphs that have a single-logarithmic approximation ratio.

翻译：本文研究需求割问题，该问题是包括多割、多路割、k-割和斯坦纳多割在内的众多经典图划分问题的推广。给定一个带边权重的图、终端组$(S_1, ..., S_g)$和整数需求$(r_1,... , r_g)$，目标是计算一个最小成本的边割，使得每个组$S_i$被分离为至少$r_i$个连通分量。尽管已有诸多研究，当前需求割问题的最佳近似算法仅能达到双对数级别的$O(\log(g).\log(n))$近似比，其方法依赖于将一般图嵌入树结构并求解树实例。本文通过探索两个尚未被充分研究的结构参数来获得单对数级别的近似比：(1) 实例中最小斯坦纳树的数量（该数量上界为图的生成树数量乘以$g$），(2) 串并联图的深度。具体而言，我们证明：若最小斯坦纳树数量为$n$的多项式函数，则简单的线性规划舍入算法可达到$O(\log n)$近似比；若图为恒定深度的串并联图，则通过对已知概率嵌入方法的精细分析，可在有界深度的串并联图上实现$O(depth.\log(g))$近似比。两项结果共同扩展了具有单对数近似比的已知图类范围。