Learning-Augmented Maximum Flow

TL;DR

This paper introduces a learning-augmented algorithm for maximum flow that leverages predictions to significantly speed up computation, achieving an $O(m ext{eta})$ runtime based on prediction accuracy, and demonstrates how to learn such predictions efficiently.

Contribution

It presents a novel framework combining machine learning predictions with maximum flow algorithms to improve runtime, and provides methods to learn optimal predictions from data.

Findings

Algorithm computes max flow in $O(m ext{eta})$ time using predictions.

Efficient PAC-learning method for minimizing prediction error.

First to improve offline problem runtime using learning predictions.

Abstract

We propose a framework for speeding up maximum flow computation by using predictions. A prediction is a flow, i.e., an assignment of non-negative flow values to edges, which satisfies the flow conservation property, but does not necessarily respect the edge capacities of the actual instance (since these were unknown at the time of learning). We present an algorithm that, given an $m$-edge flow network and a predicted flow, computes a maximum flow in $O(m\eta)$ time, where $\eta$ is the $\ell_1$ error of the prediction, i.e., the sum over the edges of the absolute difference between the predicted and optimal flow values. Moreover, we prove that, given an oracle access to a distribution over flow networks, it is possible to efficiently PAC-learn a prediction minimizing the expected $\ell_1$ error over that distribution. Our results fit into the recent line of research on learning-augmented algorithms, which aims to improve over worst-case bounds of classical algorithms by using predictions, e.g., machine-learned from previous similar instances. So far, the main focus in this area was on improving competitive ratios for online problems. Following Dinitz et al. (NeurIPS 2021), our results are one of the firsts to improve the running time of an offline problem.