Warm-starting Push-Relabel

TL;DR

This paper addresses warm-starting Push-Relabel for max-flow by leveraging a predicted pseudo-flow $\\widehat{f}$ with error $\eta$, and provides the first rigorous guarantees for runtime improvements in this setting.It introduces a three-phase algorithm that (i) converts the prediction into a cut-saturating pseudo-flow, (ii) solves excess/deficit separation across the cut via auxiliary graphs to obtain a min-cut, and (iii) completes to a max-flow, with overall time $O(\eta \cdot n^2)$ using the gap relabeling heuristic.The approach extends to general pseudo-flows that are not cut-saturating or feasible, by constructing a cut-saturating pseudo-flow first and then resolving remaining imbalances; the results are complemented by empirical validation on image segmentation tasks showcasing practical speedups for larger graphs.Overall, the work deepens understanding of how predictions can accelerate fundamental graph-optimization routines and highlights gap relabeling as a theoretically justified mechanism for improving streaming and warm-start performance in max-flow/min-cut problems.

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.

Figures (7)

• Figure 1: An illustration of different phases of warm-start Push-Relabel, starting with a cut-saturating pseudo-flow. The red curve denotes the cut. The black arrows denote the existing flows, whereas the red arrow denotes the flows sent in each phase to resolve excesses/deficits. Notice that as flows are sent, new edges become saturated and smaller cuts are found, swapping excess and deficit nodes to the opposite sides of the cut.
• Figure 2: The cropped and gray-scaled images from Figure \ref{['fig:original_images']} (copy from Figure 2 in davies23b).
• Figure 3: Cuts (red) on images chronologically evolving from the $240 \times 240$ pixel images from Birdhouse.
• Figure 4: Instances of images from each group (copy of Figure 1 from davies23b).
• Figure 5: Cuts (red) on images chronologically evolving from the $240 \times 240$ pixel images from Dog.

Theorems & Definitions (23)

• Definition 1
• Lemma 1
• Lemma 2
• Lemma 3
• proof
• Theorem 1
• Lemma 4
• proof
• Theorem 2
• proof