  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
$O(\log(\lambda^{-1} n))$.
