Fault-Tolerant Bounded Flow Preservers

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