COMBHelper: A Neural Approach to Reduce Search Space for Graph Combinatorial Problems

TL;DR

This work tackles graph-based combinatorial optimization problems that are NP-hard and plagued by large search spaces. It introduces COMBHelper, a Graph Neural Network–based pruning framework that identifies promising nodes to prune the search space, augmented by knowledge distillation and a problem-specific boosting module. The pruned space is then processed by traditional CO algorithms (LP, Greedy, Local Search) to obtain final solutions. Empirical results on synthetic and real-world graphs show at least 2× speedups while maintaining or improving solution quality across MVC and MIS problems.

Abstract

Combinatorial Optimization (CO) problems over graphs appear routinely in many applications such as in optimizing traffic, viral marketing in social networks, and matching for job allocation. Due to their combinatorial nature, these problems are often NP-hard. Existing approximation algorithms and heuristics rely on the search space to find the solutions and become time-consuming when this space is large. In this paper, we design a neural method called COMBHelper to reduce this space and thus improve the efficiency of the traditional CO algorithms based on node selection. Specifically, it employs a Graph Neural Network (GNN) to identify promising nodes for the solution set. This pruned search space is then fed to the traditional CO algorithms. COMBHelper also uses a Knowledge Distillation (KD) module and a problem-specific boosting module to bring further efficiency and efficacy. Our extensive experiments show that the traditional CO algorithms with COMBHelper are at least 2 times faster than their original versions.

Figures (6)

• Figure 1: Overview of the proposed COMBHelper. In phase 1, we generate the ground truth node labels with traditional CO algorithms on a small graph and train a teacher GCN (GCN$_{t}$) in a supervised manner; In phase 2, we train a student GCN (GCN$_{s}$) with two modules: (1) the KD module which transfers the knowledge from GCN$_{t}$ to GCN$_{s}$ and (2) the problem-specific boosting module which is used to amplify the weights of the nodes misclassified by GCN$_{t}$; In phase 3, we use GCN$_{s}$ to prune the search space of traditional CO algorithms on a large graph, i.e., predict good nodes (nodes predicted as candidate elements in the solution). Then we perform traditional CO algorithms on the reduced search space to obtain the final solution.
• Figure 2: Running times (seconds) on synthetic datasets. Our method takes less time to generate solutions for both the MVC and MIS problems. Note that all the running times of general LP (blue line) in Figure \ref{['subfig:mvclpsynthetic']} and \ref{['subfig:mislpsynthetic']} reach the time limit of 1 hour.
• Figure 3: Average speed-up on both synthetic and real-world datasets. Baselines (LP, GD, and LS) with COMBHelper$_{pt}$ and COMBHelper are at least 2 times faster than their original versions.
• Figure 4: Running times (seconds) on real-world datasets. Our method generates solutions for both the MVC and MIS problems more efficiently.
• Figure 5: Inference time (milliseconds) on synthetic and real-world datasets. GCN$_{s}$ costs less time than GCN$_{t}$ in all the cases.