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

The MAXIMUM RESIDUAL FLOW PROBLEM with one-arc destruction is posed as follows [1]. Let G = (V, A, c) be a directed network with node set V , arc setA = {e1, · · · , em}, and capacities on arcs: A 7→ R+, whereR+ 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 inG. Let aij = 1 if arc ei ∈ A lies on pathpj ∈ P andaij = 0 otherwise. Letf : P 7→ R+ be a flow (in the arc-chain form) froms to t with flow valueFf = ∑k i=1 f(pi) that satisfies capacity constraints on the arcs—that is: