# The Complexity of Computing a Robust Flow

## TL;DR

This paper investigates the computational complexity of the Maximum Robust Flow Problem, revealing new hardness results and correcting previous assumptions, with implications for network reliability analysis.

## Contribution

The authors identify a flaw in prior NP-hardness proofs, establish strong NP-hardness for unbounded k, and provide new complexity results for specific capacity cases.

## Key findings

- The problem is strongly NP-hard for unbounded k, even with limited capacity values.
- Computing optimal integral solutions is NP-hard for k=2.
- Efficient algorithms exist for capacities in {1, 2}.

## Abstract

Robust network flows are a concept for dealing with uncertainty and unforeseen failures in the network infrastructure. They and their dual counterpart, network flow interdiction, have received steady attention within the operations research community over the past years. 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 almost a decade ago: Aneja et al. (2001) showed that the problem can be solved efficiently when k = 1, while an article by Du and Chandrasekaran (2007) 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 that the problem is strongly NP-hard, even when only two different capacity values occur and the number of paths is polynomial in the size of the input. We further show that computing optimal integral solutions is already NP-hard for k = 2 (whereas for k=1, an efficient algorithm is known) and give a positive result for the case that capacities are in {1, 2}.