The maximum residual flow problem: NP-hardness with two-arc destruction

(V,A, c) be a directed network with node set V , arc set A = {e1, · · · , em}, and capacities on arcs c : A 7→ R+, where R+ is the set of non-negative rational numbers. Let s, t ∈ V be two distinct nodes, designated as the source and the destination. Let P = {p1, · · · , pk} be the set of s-t paths in G. Let aij = 1 if arc ei ∈ A lies on path pj ∈ P and aij = 0 otherwise. Let f : P 7→ R+ be a flow (in the arc-chain form) from s to t with flow value Ff = ∑k i=1 f(pi) that satisfies capacity constraints on the arcs—that is: