We introduce and discuss the Minimum Capacity-Preserving Subgraph (MCPS) problem: given a directed graph and a retention ratio $\alpha \in (0,1)$, find the smallest subgraph that, for each pair of vertices $(u,v)$, preserves at least a fraction $\alpha$ of a maximum $u$-$v$-flow's value. This problem originates from the practical setting of reducing the power consumption in a computer network: it models turning off as many links as possible while retaining the ability to transmit at least $\alpha$ times the traffic compared to the original network. First we prove that MCPS is NP-hard already on directed acyclic graphs (DAGs). Our reduction also shows that a closely related problem (which only considers the arguably most complicated core of the problem in the objective function) is NP-hard to approximate within a sublogarithmic factor already on DAGs. In terms of positive results, we present a simple linear time algorithm that solves MCPS optimally on directed series-parallel graphs (DSPs). Further, we introduce the family of laminar series-parallel graphs (LSPs), a generalization of DSPs that also includes cyclic and very dense graphs. Not only are we able to solve MCPS on LSPs in quadratic time, but our approach also yields straightforward quadratic time algorithms for several related problems such as Minimum Equivalent Digraph and Directed Hamiltonian Cycle on LSPs.

翻译：我们引入并讨论了最小保容量子图（MCPS）问题：给定一个有向图和保留比率 $\alpha \in (0,1)$，寻找一个最小的子图，使得对于每一对顶点 $(u,v)$，该子图至少保留了最大 $u$-$v$-流值的 $\alpha$ 比例。这个问题源于降低计算机网络功耗的实际场景：它模拟了在保证能传输至少原网络 $\alpha$ 倍流量的前提下，尽可能关闭更多链路的操作。首先，我们证明即使在有向无环图（DAG）上，MCPS问题也是NP难的。我们的归约还表明，一个密切相关的问题（该问题在目标函数中仅考虑问题中可以说最复杂的核心部分）即使在DAG上也是NP难的，且难以在子对数因子内近似。在正面结果方面，我们提出了一种简单的线性时间算法，该算法能在有向串并联图（DSP）上最优地求解MCPS。此外，我们引入了层状串并联图（LSP）族，这是DSP的推广，也包含循环图和高密度图。我们不仅能在LSP上以二次时间求解MCPS，而且我们的方法还能为LSP上的多个相关问题（如最小等价有向图和哈密顿有向环）直接给出二次时间算法。