Robust static and dynamic maximum flows

TL;DR

This paper introduces a unifying framework for robust maximum flow problems, analyzing their complexity and demonstrating that the new models outperform existing ones in solution quality.

Contribution

It presents a novel general model for robust maximum flows that encompasses existing models, analyzes their computational complexity, and compares solution qualities.

Findings

General models are NP-hard, but static case with Γ=1 is polynomially solvable.

The new models yield better robust solutions than existing models.

Bounds on the solution quality gaps are established.

Abstract

We study the robust maximum flow problem and the robust maximum flow over time problem where a given number of arcs $\Gamma$ may fail or may be delayed. Two prominent models have been introduced for these problems: either one assigns flow to arcs fulfilling weak flow conservation in any scenario, or one assigns flow to paths where an arc failure or delay affects a whole path. We provide a unifying framework by presenting novel general models, in which we assign flow to subpaths. These models contain the known models as special cases and unify their advantages in order to obtain less conservative robust solutions. We give a thorough analysis with respect to complexity of the general models. In particular, we show that the general models are essentially NP-hard, whereas, e.g. in the static case with $\Gamma = 1$ an optimal solution can be computed in polynomial time. Further, we answer the open question about the complexity of the dynamic path model for $\Gamma = 1$. We also compare the solution quality of the different models. In detail, we show that the general models have better robust optimal values than the known models and we prove bounds on these gaps.