Taming Treewidth DP with Modulators: A General Booster for Graph Heuristics

2026-08-05

AI Summary

Traditional tree decomposition–based dynamic programming struggles to scale to large graphs, limiting its applicability to NP-hard graph optimization problems. This work proposes a general-purpose enhancement framework that overcomes this limitation by efficiently constructing treewidth modulators to partition the original graph into an easily solvable component and a residual subgraph of small treewidth. The framework seamlessly integrates heuristic methods—such as greedy algorithms, evolutionary search, and graph neural networks—with tree decomposition–based dynamic programming in a synergistic manner. Notably, it introduces tree decomposition–based dynamic programming as a universal booster applicable across diverse heuristic paradigms. Empirical evaluations on the maximum independent set, minimum vertex cover, and maximum cut problems demonstrate substantial performance gains over baseline algorithms; remarkably, the enhanced greedy approach matches or even surpasses state-of-the-art commercial solvers.

Abstract

Treewidth is a fundamental graph invariant that quantifies how tree-like a given graph is. It is extensively used with dynamic programming to design fixed-parameter tractable algorithms for many NP-hard graph combinatorial optimization problems. However, despite broad theoretical applicability, treewidth dynamic programming (TDP) does not scale in practice beyond graphs with very small treewidth. Rather than applying TDP as a standalone technique, in this paper, we demonstrate that TDP can serve as a broadly applicable enhancer for a wide range of graph combinatorial optimization algorithms. Our framework leverages the concept of treewidth modulators, which refer to vertex sets whose removal significantly reduces the treewidth. We further propose an empirically efficient procedure for generating such treewidth modulators. To enhance an algorithm $\textit{A}$, we use $\textit{A}$ to heuristically make decisions on the modulators vertices, after which the remaining decisions outside the treewidth modulators become scalable for TDP. To demonstrate the general applicability of our proposed framework. We experimented with three classic graph combinatorial optimization models: Maximum Independent Set, Minimum Vertex Cover, and Max Cut. We apply TDP to enhance algorithms across diverse paradigms, including evolutionary search, greedy heuristics, and graph-neural-network-based heuristics. For all combinations of optimization models and base algorithms, TDP significantly improves performance over the original methods. In many settings, TDP-enhanced greedy heuristics are competitive with, and sometimes clearly outperform, state-of-the-art commercial solvers.

Problem

• treewidth
• dynamic programming
• graph combinatorial optimization
• modulators
• scalability

Innovation

• treewidth modulator
• dynamic programming
• graph combinatorial optimization
• fixed-parameter tractability
• heuristic enhancement