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Faster Global Minimum Cut with Predictions

Benjamin Moseley, Helia Niaparast, Karan Singh

TL;DR

This work shows how machine-learned predictions can dramatically accelerate the global minimum cut problem by warm-starting randomized edge-contraction algorithms. It introduces prediction-augmented variants of Karger's algorithm and the Karger-Stein/FPZ approach, parameterized by asymmetric error metrics η and ρ that quantify prediction quality, and proves substantial runtime improvements with provable probabilistic guarantees. A learning framework demonstrates how near-optimal predictions can be learned from past instances by convexifying a surrogate objective and applying online gradient methods, with guarantees that the learned predictor approaches the best expected runtime over a distribution. Empirical results across synthetic, TSP-LP-derived, and real graphs confirm large speedups and robustness to prediction errors, highlighting practical potential for speeding up repeated minimum-cut computations in related-instance settings.

Abstract

Global minimum cut is a fundamental combinatorial optimization problem with wide-ranging applications. Often in practice, these problems are solved repeatedly on families of similar or related instances. However, the de facto algorithmic approach is to solve each instance of the problem from scratch discarding information from prior instances. In this paper, we consider how predictions informed by prior instances can be used to warm-start practical minimum cut algorithms. The paper considers the widely used Karger's algorithm and its counterpart, the Karger-Stein algorithm. Given good predictions, we show these algorithms become near-linear time and have robust performance to erroneous predictions. Both of these algorithms are randomized edge-contraction algorithms. Our natural idea is to probabilistically prioritize the contraction of edges that are unlikely to be in the minimum cut.

Faster Global Minimum Cut with Predictions

TL;DR

This work shows how machine-learned predictions can dramatically accelerate the global minimum cut problem by warm-starting randomized edge-contraction algorithms. It introduces prediction-augmented variants of Karger's algorithm and the Karger-Stein/FPZ approach, parameterized by asymmetric error metrics η and ρ that quantify prediction quality, and proves substantial runtime improvements with provable probabilistic guarantees. A learning framework demonstrates how near-optimal predictions can be learned from past instances by convexifying a surrogate objective and applying online gradient methods, with guarantees that the learned predictor approaches the best expected runtime over a distribution. Empirical results across synthetic, TSP-LP-derived, and real graphs confirm large speedups and robustness to prediction errors, highlighting practical potential for speeding up repeated minimum-cut computations in related-instance settings.

Abstract

Global minimum cut is a fundamental combinatorial optimization problem with wide-ranging applications. Often in practice, these problems are solved repeatedly on families of similar or related instances. However, the de facto algorithmic approach is to solve each instance of the problem from scratch discarding information from prior instances. In this paper, we consider how predictions informed by prior instances can be used to warm-start practical minimum cut algorithms. The paper considers the widely used Karger's algorithm and its counterpart, the Karger-Stein algorithm. Given good predictions, we show these algorithms become near-linear time and have robust performance to erroneous predictions. Both of these algorithms are randomized edge-contraction algorithms. Our natural idea is to probabilistically prioritize the contraction of edges that are unlikely to be in the minimum cut.

Paper Structure

This paper contains 20 sections, 11 theorems, 39 equations, 2 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Global Minimum Cut.
  3. Karger's and Karger-Stein Algorithms.
  4. Results
  5. Predictions.
  6. Boosted Karger's Algorithm.
  7. Boosting the Karger-Stein Algorithm.
  8. Learning Near-Optimal Predictions.
  9. Experiments.
  10. Limitations.
  11. Boosted Karger's Algorithm
  12. Boosting the Karger-Stein Algorithm
  13. Learning Near-Optimal Predictions
  14. Experiments
  15. Controlled Experiments on Synthetic Graphs
  16. ...and 5 more sections

Key Result

Theorem 1.1

For a suitable setting of parameters, given predictions measured by $\eta$ and $\rho$ as defined above, the Boosted Karger's algorithm (alg:boosted) outputs a minimum cut with probability at least

Figures (2)

  • Figure 1: A controlled experimental comparison of the number of repetitions Boosted Karger's algorithm needs to find the mincut vs. the standard Karger's algorithm for different quality of predictions, as parameterized by $\eta$ and $\rho$.
  • Figure 2: In \ref{['fig:TSP']}A, on the left, we compare the number of minimum cuts arising from the subtour TSP that Karger's and Boosted Karger's algorithms solved within a given number of repetitions. On the right, in \ref{['fig:TSP']}B, is demonstrated the number of repetitions needed to recover the minimum cut on three real graph datasets.

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 3.1
  • proof
  • ...and 9 more