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Preference-Based Gradient Estimation for ML-Guided Approximate Combinatorial Optimization

Arman Mielke, Uwe Bauknecht, Thilo Strauss, Mathias Niepert

TL;DR

This work tackles the challenge of improving combinatorial optimization (CO) solutions within time budgets by marrying fast, non-learned approximation algorithms with graph neural networks (GNNs) that predict input parameters. It introduces Preference-Based Gradient Estimation (PBGE), a self-supervised method to backpropagate through non-differentiable solvers by comparing solution pairs sampled from the solver and ranking them by the CO objective. The approach yields a provably effective gradient signal, scales to problems like minimum $k$-cut and TSP, and achieves strong performance on standard benchmarks without requiring ground-truth labels. Practically, PBGE enables efficient, data-driven guidance of classic CO heuristics, improving solution quality while maintaining fast runtimes and broad applicability to graph-based CO tasks.

Abstract

Combinatorial optimization (CO) problems arise across a broad spectrum of domains, including medicine, logistics, and manufacturing. While exact solutions are often computationally infeasible, many practical applications require high-quality solutions within a given time budget. To address this, we propose a learning-based approach that enhances existing non-learned approximation algorithms for CO. Specifically, we parameterize these approximation algorithms and train graph neural networks (GNNs) to predict parameter values that yield near-optimal solutions. Our method is trained end-to-end in a self-supervised fashion, using a novel gradient estimation scheme that treats the approximation algorithm as a black box. This approach combines the strengths of learning and traditional algorithms: the GNN learns from data to guide the algorithm toward better solutions, while the approximation algorithm ensures feasibility. We validate our method on two well-known combinatorial optimization problems: the travelling salesman problem (TSP) and the minimum k-cut problem. Our results demonstrate that the proposed approach is competitive with state-of-the-art learned CO solvers.

Preference-Based Gradient Estimation for ML-Guided Approximate Combinatorial Optimization

TL;DR

This work tackles the challenge of improving combinatorial optimization (CO) solutions within time budgets by marrying fast, non-learned approximation algorithms with graph neural networks (GNNs) that predict input parameters. It introduces Preference-Based Gradient Estimation (PBGE), a self-supervised method to backpropagate through non-differentiable solvers by comparing solution pairs sampled from the solver and ranking them by the CO objective. The approach yields a provably effective gradient signal, scales to problems like minimum -cut and TSP, and achieves strong performance on standard benchmarks without requiring ground-truth labels. Practically, PBGE enables efficient, data-driven guidance of classic CO heuristics, improving solution quality while maintaining fast runtimes and broad applicability to graph-based CO tasks.

Abstract

Combinatorial optimization (CO) problems arise across a broad spectrum of domains, including medicine, logistics, and manufacturing. While exact solutions are often computationally infeasible, many practical applications require high-quality solutions within a given time budget. To address this, we propose a learning-based approach that enhances existing non-learned approximation algorithms for CO. Specifically, we parameterize these approximation algorithms and train graph neural networks (GNNs) to predict parameter values that yield near-optimal solutions. Our method is trained end-to-end in a self-supervised fashion, using a novel gradient estimation scheme that treats the approximation algorithm as a black box. This approach combines the strengths of learning and traditional algorithms: the GNN learns from data to guide the algorithm toward better solutions, while the approximation algorithm ensures feasibility. We validate our method on two well-known combinatorial optimization problems: the travelling salesman problem (TSP) and the minimum k-cut problem. Our results demonstrate that the proposed approach is competitive with state-of-the-art learned CO solvers.

Paper Structure

This paper contains 51 sections, 6 theorems, 37 equations, 4 figures, 5 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Background
  4. Combinatorial optimization problems.
  5. Graph neural networks (GNNs).
  6. Gradient Estimators for Discrete Distributions.
  7. Problem statement
  8. Method
  9. Parameterizing approximation algorithms
  10. Preference-based gradient estimation (PBGE)
  11. Gradient scaling.
  12. Theoretical analysis
  13. Experiments
  14. Problem instance generation and baselines
  15. Results
  16. ...and 36 more sections

Key Result

Theorem 1

Let $h(\hat{\bm{y}} \mid {\mathcal{G}}, \bm{s})$ be the Karger--Stein algorithm guided by scores $\bm{s} \in \mathbb{R}^{|E|}$, and let $\bm{y} \in \{0, 1\}^{|E|}$ be a minimum $k$-cut on graph ${\mathcal{G}} = (V, E)$. Then,

Figures (4)

  • Figure 1: Overview of the proposed framework for ML-guided approximate combinatorial optimization. In the forward pass, the GNN predicts edge scores that are used as input parameters for a CO approximation algorithm. The approximation algorithm operates on these parameters by using them to scale the graph's edge weights and running an off-the-shelf approximation algorithm on the modified graph to obtain a solution $\hat{\bm{y}}$. Since the CO approximation algorithm is not differentiable in general, a gradient estimation scheme is used in the backward pass.
  • Figure 2: The number of graphs for which the minimum $k$-cut was found after a given number of Karger--Stein runs. For example, for Karger--Stein without a GNN on graphs with 100 nodes (\ref{['fig:karger-stein-num-runs-until-minimum-found-n-100']}), two Karger--Stein runs suffice to find the minimum $k$-cut for 83% of graphs.
  • Figure 3: The Pareto frontier of self-supervised methods on TSP-100. Non-Pareto-optimal baselines with very high optimality gaps or very long runtimes are omitted for clarity.
  • Figure : One training iteration with PBGE

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 3