Given a directed graph $G = (V, E)$ with $n$ vertices, $m$ edges and a designated source vertex $s\in V$, we consider the question of finding a sparse subgraph $H$ of $G$ that preserves the flow from $s$ up to a given threshold $\lambda$ even after failure of $k$ edges. We refer to such subgraphs as $(\lambda,k)$-fault-tolerant bounded-flow-preserver ($(\lambda,k)$-FT-BFP). Formally, for any $F \subseteq E$ of at most $k$ edges and any $v\in V$, the $(s, v)$-max-flow in $H \setminus F$ is equal to $(s, v)$-max-flow in $G \setminus F$, if the latter is bounded by $\lambda$, and at least $\lambda$ otherwise. Our contributions are summarized as follows: 1. We provide a polynomial time algorithm that given any graph $G$ constructs a $(\lambda,k)$-FT-BFP of $G$ with at most $\lambda 2^kn$ edges. 2. We also prove a matching lower bound of $\Omega(\lambda 2^kn)$ on the size of $(\lambda,k)$-FT-BFP. In particular, we show that for every $\lambda,k,n\geq 1$, there exists an $n$-vertex directed graph whose optimal $(\lambda,k)$-FT-BFP contains $\Omega(\min\{2^k\lambda n,n^2\})$ edges. 3. Furthermore, we show that the problem of computing approximate $(\lambda,k)$-FT-BFP is NP-hard for any approximation ratio that is better than $O(\log(\lambda^{-1} n))$.

翻译：给定一个有向图 $G = (V, E)$，包含 $n$ 个顶点、$m$ 条边和一个指定源顶点 $s\in V$，我们研究在 $k$ 条边发生故障后，如何找到一个稀疏子图 $H \subseteq G$，使得其能保持从 $s$ 出发且不超过给定阈值 $\lambda$ 的流。我们将这类子图称为 $(\lambda,k)$-容错有界流保持子图（$(\lambda,k)$-FT-BFP）。形式化地，对于任意至多 $k$ 条边的故障集 $F \subseteq E$ 和任意 $v\in V$，若 $G \setminus F$ 中 $(s, v)$-最大流不超过 $\lambda$，则 $H \setminus F$ 中 $(s, v)$-最大流等于该值，否则至少为 $\lambda$。我们的贡献总结如下：1. 我们提出一个多项式时间算法，对于任意图 $G$，可构造一个至多包含 $\lambda 2^k n$ 条边的 $(\lambda,k)$-FT-BFP。2. 我们证明了该尺寸的下界为 $\Omega(\lambda 2^k n)$ 并与之匹配。特别地，对于任意 $\lambda,k,n\geq 1$，存在一个 $n$ 顶点的有向图，其最优 $(\lambda,k)$-FT-BFP 至少包含 $\Omega(\min\{2^k\lambda n,n^2\})$ 条边。3. 此外，我们证明计算近似 $(\lambda,k)$-FT-BFP 的问题，在近似比优于 $O(\log(\lambda^{-1} n))$ 时是 NP-难的。