Warm-starting Push-Relabel

TL;DR

This paper introduces a theoretically justified method for warm-starting the Push-Relabel algorithm with predicted flows, improving efficiency when predictions are accurate and maintaining robustness in worst-case scenarios.

Contribution

It provides the first theoretical guarantees for warm-starting Push-Relabel with predictions and explains the effectiveness of the gap relabeling heuristic.

Findings

The algorithm benefits from fast running time with accurate predictions.

It maintains robust worst-case guarantees.

Experimental results confirm practical effectiveness.

Abstract

Push-Relabel is one of the most celebrated network flow algorithms. Maintaining a pre-flow that saturates a cut, it enjoys better theoretical and empirical running time than other flow algorithms, such as Ford-Fulkerson. In practice, Push-Relabel is even faster than what theoretical guarantees can promise, in part because of the use of good heuristics for seeding and updating the iterative algorithm. However, it remains unclear how to run Push-Relabel on an arbitrary initialization that is not necessarily a pre-flow or cut-saturating. We provide the first theoretical guarantees for warm-starting Push-Relabel with a predicted flow, where our learning-augmented version benefits from fast running time when the predicted flow is close to an optimal flow, while maintaining robust worst-case guarantees. Interestingly, our algorithm uses the gap relabeling heuristic, which has long been employed in practice, even though prior to our work there was no rigorous theoretical justification for why it can lead to run-time improvements. We then provide experiments that show our warm-started Push-Relabel also works well in practice.