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Neural Tractability via Structure: Learning-Augmented Algorithms for Graph Combinatorial Optimization

Jialiang Li, Weitong Chen, Mingyu Guo

TL;DR

This work tackles NP-hard graph combinatorial optimization by merging fast neural inference with structure-aware exact search. By partitioning instances into a treewidth-based hard region and an easy region, the neural advisor fixes decisions in the hard region while a treewidth-constrained dynamic program (TDPA) solves the rest exactly, yielding higher-quality solutions with strong generalization. The proposed neural fixed-parameter tractable framework (N-FPT) is model-agnostic and demonstrates substantial improvements over pure neural solvers and competitive performance against commercial solvers across MIS, MVC, and MC. The approach offers practical scalability and robustness to distribution shifts, highlighting the value of combining learning with parameterized algorithmic structure for graph CO.

Abstract

Neural models have shown promise in solving NP-hard graph combinatorial optimization (CO) problems. Once trained, they offer fast inference and reasonably high-quality solutions for in-distribution testing instances, but they generally fall short in terms of absolute solution quality compared to classical search-based algorithms that are admittedly slower but offer optimality guarantee once search finishes. We propose a novel framework that combines the inference efficiency and exploratory power of neural models with the solution quality guarantee of search-based algorithms. In particular, we use parameterized algorithms (PAs) as the search component. PAs are dedicated to identifying easy instances of generally NP-hard problems, and allow for practically efficient search by exploiting structural simplicity (of the identified easy instances). Under our framework, we use parameterized analysis to identify the structurally hard parts of a CO instance. The neural model handles the hard parts by generating advisory signals based on its data-driven understanding. The PA-based search component then integrates the advisory signals to systematically and efficiently searches through the remaining structurally easy parts. Notably, our framework is agnostic to the choice of neural model and produces strictly better solutions than neural solvers alone. We examine our framework on multiple CO tasks. Empirical results show that it achieves superior solution quality, competitive with that of commercial solvers. Furthermore, by using the neural model only for exploratory advisory signals, our framework exhibits improved out-of-distribution generalization, addressing a key limitation of existing neural CO solvers.

Neural Tractability via Structure: Learning-Augmented Algorithms for Graph Combinatorial Optimization

TL;DR

This work tackles NP-hard graph combinatorial optimization by merging fast neural inference with structure-aware exact search. By partitioning instances into a treewidth-based hard region and an easy region, the neural advisor fixes decisions in the hard region while a treewidth-constrained dynamic program (TDPA) solves the rest exactly, yielding higher-quality solutions with strong generalization. The proposed neural fixed-parameter tractable framework (N-FPT) is model-agnostic and demonstrates substantial improvements over pure neural solvers and competitive performance against commercial solvers across MIS, MVC, and MC. The approach offers practical scalability and robustness to distribution shifts, highlighting the value of combining learning with parameterized algorithmic structure for graph CO.

Abstract

Neural models have shown promise in solving NP-hard graph combinatorial optimization (CO) problems. Once trained, they offer fast inference and reasonably high-quality solutions for in-distribution testing instances, but they generally fall short in terms of absolute solution quality compared to classical search-based algorithms that are admittedly slower but offer optimality guarantee once search finishes. We propose a novel framework that combines the inference efficiency and exploratory power of neural models with the solution quality guarantee of search-based algorithms. In particular, we use parameterized algorithms (PAs) as the search component. PAs are dedicated to identifying easy instances of generally NP-hard problems, and allow for practically efficient search by exploiting structural simplicity (of the identified easy instances). Under our framework, we use parameterized analysis to identify the structurally hard parts of a CO instance. The neural model handles the hard parts by generating advisory signals based on its data-driven understanding. The PA-based search component then integrates the advisory signals to systematically and efficiently searches through the remaining structurally easy parts. Notably, our framework is agnostic to the choice of neural model and produces strictly better solutions than neural solvers alone. We examine our framework on multiple CO tasks. Empirical results show that it achieves superior solution quality, competitive with that of commercial solvers. Furthermore, by using the neural model only for exploratory advisory signals, our framework exhibits improved out-of-distribution generalization, addressing a key limitation of existing neural CO solvers.

Paper Structure

This paper contains 34 sections, 1 theorem, 7 equations, 9 figures, 9 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Model and Notation.
  4. Learning-Augmented Algorithms (LAs).
  5. Proposed Method
  6. Fixed-Parameter Tractability (FPT).
  7. Courcelle's Theorem Courcelle1990TheMS.
  8. Tree Decompositions (TD) & Treewidth (TW) Robertson1984GraphMI.
  9. Treewidth Modulator (TM) cygan2014hardness.
  10. Neural Fixed-Parameter Tractable Algorithm (N-FPT) (Figure \ref{['fig:arch']}).
  11. Experiments
  12. Configurations.
  13. Results & Analysis.
  14. Average sampling Improvements
  15. Best-of-N Improvements
  16. ...and 19 more sections

Key Result

proposition 1

Let $\mathtt{Orc}_{\theta}$ be the neural solver behind N-FPT. We assume an (maximization) objective function $f$, a graph $G$, a target treewidth $\eta$, and the corresponding treewidth modulator $TM_{\eta}$. Let $y\coloneq \mathcal{G}\to 2^V$ be a perfect oracle offering the optimal vertex selecti

Figures (9)

  • Figure 1: Tree decomposition and treewidth modulator: the left image shows the original TD, where the largest bag has seven nodes, whose treewidth is 6. Given a target treewidth $\eta=4$, vertex deletions are required. Blue $\bullet$ and red $\bullet$ highlight a valid (not necessarily optimal) treewidth modulator.
  • Figure 2: N-FPT overview: The vertex set $V$ is split into TM and $V\setminus\textrm{TM}$. We query GFlowNet for decisions in TM, which are injected to TDPA. In the rightmost TD, "?" denotes undecided vertices in $V\setminus\textrm{TM}$, and only these undecided vertices will be enumerated during DP.
  • Figure 3: Incremental confidence level: the process begins with a fully undecided graph state $\left\{?\right\}^{\left|V\right|}$. GFlowNet generates multiple samples, from which consensus decisions via majority voting are committed. The resulting state then seeds the next rollout.
  • Figure 4: Randomized deferral: the process starts from a fully undecided graph state $\left\{?\right\}^{\left|V\right|}$. Once a trajectory is complete, some decided vertices are randomly reverted back to '?'; if an 1-vertex is reverted, its neighbors are also reset to '?'. The resulting reverted state then seeds the next rollout.
  • Figure 5: Results for intra-class generalization. We report the relative performance using $\frac{B}{A}-1$, where $A$ is the result for the baseline GFlowNet, and $B$ is the result of compound methods, as specified by the legend. Higher values indicate greater gains, emphasizing improved generalization.
  • ...and 4 more figures

Theorems & Definitions (1)

  • proposition 1