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

Abstract

The maximum residual flow problem with one-arc destruction is shown to be solvable in strongly polynomial time in [Aneja et al., Networks, 38 (2001), 194-198]. However, the status of the corresponding problem with more than one-arc destruction is left open therein. We resolve the status of the two-arc destruction problem by demonstrating that it is already NP -hard.

The complexity of computing a robust flow

“…One of the most basic models is the Maximum Robust Flow problem: Given a network and an integer k, the task is to find a path flow of maximum robust value, i.e., the guaranteed value of surviving flow after removal of any k arcs in the network. The complexity of this problem appeared to have been settled a decade ago: Aneja et al [1] showed that the problem can be solved efficiently when k = 1, while an article by Du and Chandrasekaran [2] established that the problem is NP -hard for any constant value of k larger than 1.We point to a flaw in the proof of the latter result, leaving the complexity for constant k open once again. For the case that k is not bounded by a constant, we present a new hardness proof, establishing NP -hardness even for instances where the number of paths is polynomial in the size of the network.…”

“…Du and Chandrasekaran [2] showed that Separation(Q) is NP -hard, even when k = 2. They concluded that by the equivalence of optimization and separation, solving [D] and hence solving [P] is NP-hard.…”

Reduction of Maximum Flow Network Interdiction Problem: Step towards the Polynomial Time Solutions

“…The decomposition method had been used in that algorithm that either returned a After that less attention were paid to develop the linear program and approximation algorithms for MFNIP and much of the work included the study of variants of network interdiction problem ( [10], [11], [12], [17], [18], [20] , [12] , [31]). …”

Development of an Algorithm for all type of Network Flow Problems

“…A similar problem to MFNIP, called the Network Inhibition Problem (NIP), was independently introduced by Phillips in [14]. Numerous variants of network interdiction problems have also been studied, including shortest path network interdiction [10], stochastic network interdiction [7] and [11], multiple commodity network interdiction [12] and [19], facility interdiction [6], and a variant of MFNIP where flow is routed before arcs are removed [8].…”