Fault-Tolerant Bounded Flow Preservers

Abstract

Given a directed graph G = (V, E) with n vertices, m edges and a designated source vertex s ∈ V, we consider the question of finding a sparse subgraph H of G that preserves the flow from s up to a given threshold λ even after failure of k edges. We refer to such subgraphs as (λ,k)-fault-tolerant bounded-flow-preserver ((λ,k)-FT-BFP). Formally, for any F ⊆ E of at most k edges and any v ∈ V, the (s, v)-max-flow in H⧵F is equal to (s, v)-max-flow in G⧵F, if the latter is bounded by λ, and at least λ otherwise. Our contributions are summarized as follows:
1) We provide a polynomial time algorithm that given any graph G constructs a (λ,k)-FT-BFP of G with at most λ 2^kn edges. 
2) We also prove a matching lower bound of Ω(λ 2^kn) on the size of (λ,k)-FT-BFP. In particular, we show that for every λ,k,n ⩾ 1, there exists an n-vertex directed graph whose optimal (λ,k)-FT-BFP contains Ω(min{2^kλ n, n²}) edges.
3) Furthermore, we show that the problem of computing approximate (λ,k)-FT-BFP is NP-hard for any approximation ratio that is better than O(log(λ^{-1} n)).