Learning-Augmented Maximum Flow
This work addresses the challenge of improving worst-case running times for offline problems in algorithm design, specifically for maximum flow, by integrating machine learning predictions, which is an incremental advance in the learning-augmented algorithms field.
The paper tackles the problem of speeding up maximum flow computation by using predicted flows, achieving an algorithm that computes a maximum flow in O(mη) time, where η is the ℓ₁ error of the prediction, and shows that predictions can be PAC-learned to minimize expected error.
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 $O(mη)$ time, where $η$ 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.