Learning Combinatorial Optimization Algorithms over Graphs (1704.01665v4)

Abstract: The design of good heuristics or approximation algorithms for NP-hard combinatorial optimization problems often requires significant specialized knowledge and trial-and-error. Can we automate this challenging, tedious process, and learn the algorithms instead? In many real-world applications, it is typically the case that the same optimization problem is solved again and again on a regular basis, maintaining the same problem structure but differing in the data. This provides an opportunity for learning heuristic algorithms that exploit the structure of such recurring problems. In this paper, we propose a unique combination of reinforcement learning and graph embedding to address this challenge. The learned greedy policy behaves like a meta-algorithm that incrementally constructs a solution, and the action is determined by the output of a graph embedding network capturing the current state of the solution. We show that our framework can be applied to a diverse range of optimization problems over graphs, and learns effective algorithms for the Minimum Vertex Cover, Maximum Cut and Traveling Salesman problems.

An Overview of "Learning Combinatorial Optimization Algorithms over Graphs"

The paper "Learning Combinatorial Optimization Algorithms over Graphs" by Hanjun Dai, Elias B. Khalil, Yuyu Zhang, Bistra Dilkina, and Le Song, addresses a crucial and challenging problem in combinatorial optimization: automating the design of heuristics for NP-hard graph problems. The paper introduces a novel framework that leverages a combination of reinforcement learning (RL) and graph embedding techniques to learn effective greedy heuristics for a variety of combinatorial optimization problems over graphs, including Minimum Vertex Cover (MVC), Maximum Cut (MAXCUT), and the Traveling Salesman Problem (TSP).

Key Contributions

1. Greedy Meta-Algorithm Design: The authors propose a greedy meta-algorithm, where a solution is incrementally constructed by adding nodes sequentially while satisfying the problem's constraints. This design is parameterized to apply across different graph problems seamlessly.
2. Graph Embedding for Policy Representation: A significant innovation in this paper is the use of a graph embedding network, named structure2vec (S2V), to represent the policy in the greedy algorithm. This allows the policy to capture rich relational information between nodes in a graph, making it generalizable to graphs of different sizes.
3. Reinforcement Learning Framework: The evaluation function within the greedy algorithm is optimized using fitted Q-learning, a variant of Q-learning tailored to handle delayed rewards effectively, which is crucial for making decisions in complex combinatorial optimization tasks.

Methodology and Implementation

1. Problem Formulation: The greedy algorithms for each of the optimization problems (MVC, MAXCUT, and TSP) are framed under a common formulation. The state of the problem is represented by the current partial solution, while the reward function is defined as the change in objective value resulting from adding a node to the partial solution.
2. Graph Embedding: The structure2vec (S2V) network iteratively updates node embeddings by aggregating information over the graph structure, ensuring that node representations encode multi-hop neighborhood information. This embedding framework effectively transforms the combinatorial structure of the graph into a rich feature space useful for learning policies.
3. Q-Learning Implementation: The authors employ an n-step Q-learning approach with experience replay to train the graph embedding parameters. This ensures sample efficiency and stable training, which is vital given the complexity and size variability of graph instances.

Experimental Evaluation

A thorough empirical evaluation demonstrates the effectiveness of the proposed framework:

• Solution Quality: S2V-DQN (Deep Q-Network using structure2vec) achieves competitive or superior approximation ratios compared to both classical heuristics and state-of-the-art learning approaches like Pointer Networks with Actor-Critic (PN-AC).
• Generalization: The framework exhibits remarkable generalization, maintaining high-quality solutions on test graphs significantly larger than those encountered during training (up to 1200 nodes).
• Running Time: Despite the complexity of the learned models, the method demonstrates practical efficiency, often requiring less time to achieve high-quality solutions compared to traditional exact methods with similar or better performance.

Implications and Future Directions

1. Automation of Algorithm Design: This work marks a significant step towards automating the design of heuristics for combinatorial optimization, reducing the need for problem-specific hand-crafting of algorithms.
2. Scalability: The framework's ability to generalize to larger instances underscores the potential for applying similar techniques to industrial-scale problems, where real-time decision support is crucial.
3. New Algorithm Discovery: Interestingly, the learned heuristics reveal strategies that are intuitive yet previously unexplored, suggesting that these methods could contribute to discovering novel algorithmic insights, particularly for less-studied optimization problems.

Concluding Remarks

The paper presents a robust method for learning greedy heuristics for combinatorial optimization problems on graphs, leveraging powerful techniques from reinforcement learning and graph neural networks. Potential future developments could involve exploring different types of graph embeddings, integrating other forms of RL like actor-critic methods, and extending the framework to more complex optimization scenarios like multi-objective problems or dynamic graph settings. The successful application to diverse problems like MVC, MAXCUT, and TSP opens promising avenues for both theoretical advancements and practical implementations in AI and graph algorithm design.