[Paper Review] Combinatorial Optimization with Graph Convolutional Networks and Guided Tree Search

Zhuwen Li, Qifeng Chen+1 more|arXiv (Cornell University)|Oct 25, 2018

Advanced Graph Neural Networks25 references216 citations

TL;DR

The paper presents a learning-based approach using a graph convolutional network to predict vertex inclusion likelihood for NP-hard problems, combined with a diversity-enabled tree search and classic heuristics to generate and refine multiple candidate solutions.

ABSTRACT

We present a learning-based approach to computing solutions for certain NP-hard problems. Our approach combines deep learning techniques with useful algorithmic elements from classic heuristics. The central component is a graph convolutional network that is trained to estimate the likelihood, for each vertex in a graph, of whether this vertex is part of the optimal solution. The network is designed and trained to synthesize a diverse set of solutions, which enables rapid exploration of the solution space via tree search. The presented approach is evaluated on four canonical NP-hard problems and five datasets, which include benchmark satisfiability problems and real social network graphs with up to a hundred thousand nodes. Experimental results demonstrate that the presented approach substantially outperforms recent deep learning work, and performs on par with highly optimized state-of-the-art heuristic solvers for some NP-hard problems. Experiments indicate that our approach generalizes across datasets, and scales to graphs that are orders of magnitude larger than those used during training.

Motivation & Objective

• Motivate solving NP-hard problems by combining deep learning with heuristic algorithmic techniques.
• Propose a graph convolutional network that predicts vertex likelihood of being in an optimal solution.
• Enable diverse solution synthesis to navigate multimodal solution spaces.
• Integrate GCN predictions with a breadth-first, parallel tree search to generate and refine multiple candidate solutions.
• Demonstrate generalization and scalability across datasets and graph sizes.

Proposed method

• Represent NP-hard problems (MIS, MVC, MC, SAT) as MIS instances and learn vertex-wise likelihoods with a graph convolutional network.
• Train a GCN f(G;θ) to output a probability map over vertices; use sigmoid on the final layer and binary cross-entropy loss for supervised learning.
• Address multimodality by generating M diverse probability maps per graph using a diverse-solution training objective (hindsight loss).
• Use the M maps to seed a breadth-first, parallel tree search that expands incomplete solutions into many complete candidates.
• Refine candidate solutions with a 2-improvement local search and apply graph reduction to speed up processing.
• Evaluate and compare against SOTA deep learning baselines, SMT/MIP solvers, and classic heuristics on SAT, MIS, MVC, and MC benchmarks.

Experimental results

Research questions

• RQ1 Can a GCN predict vertex participation likelihood for optimal solutions across multiple NP-hard problems when trained on synthetic data?
• RQ2 Does generating diverse solution maps improve exploration of the solution space and improve final solution quality?
• RQ3 How does the proposed approach compare to strong traditional solvers and recent deep learning methods on benchmarks of varying size and structure?
• RQ4 Does the method generalize across datasets and scale to graphs with up to ~100k nodes?
• RQ5 What is the impact of design choices (diversity loss, tree search strategy, and local search) on performance?

Key findings

• The approach solves all SATLIB test instances and matches state-of-the-art SMT/MIS solvers on SAT benchmarks.
• On SAT Competition 2017, the method solves 100% of instances, outperforming several baselines.
• On BUAA-MC, the method solves 62.5% of MC instances, and outperforms baselines on MIS and MVC.
• Across SNAP social networks and citation graphs, the method outperforms most baselines and matches or exceeds ReduMIS on MIS and MVC.
• The model generalizes from synthetic SAT graphs to real-world networks and scales to graphs an order of magnitude larger than training data.