Learning to solve graph metric dimension problem based on graph contrastive learning

Abstract

Deep learning has been widely used to solve graph and combinatorial optimization problems. However, proper model deployment is critical for training a model and solving all problems. Existing frameworks mainly use reinforcement learning to learn to solve combinatorial optimization problems, in which a partial solution of the problem is regarded as an environmental state and each vertex of the corresponding graph is regarded as an action. As a result, using the sample data in model training effectively is challenging for different graphs. This study proposes a sampling-based, data-driven and distributed independent graph learning framework, based on decoupling graph structure learning and problem solving processes. To some extent, it facilitates industrial applications. Specifically, the framework consists of two independent parts: extracting graph structure and learning to solve the problem. Under this framework, the graph contrastive learning(GCL) is used to finish the graph structure learning process. Then by means of state-value aggregation on all of nodes in graphs, a global reinforcement learning method is established to learn to solve the graph problem, associated with repair policies to get improvement of performance. Experiments on synthetic graph datasets show that the graph contrastive learning is beneficial or has some advantages for training stability and improving the accuracy of solving the graph problem, and that the repair policies are stable for solution search. However, it also demonstrates that the graph neural network is not necessarily needed in the process of learning to solve the graph problem. Moreover, learning to solve MDP still has some challenges, such as decreasing learning performance with increasing edge existence probability of graphs, and it is unknown what kind of reward function is appropriate for solving MDP.

Appendices

Appendix A: model setup

1.1 A.1 GCN-NF model

The GNN-NF model consist of one graph convolutional network(GCN) layer and a projection layer. Moreover, the projection layer is a two-layer perceptron. Specifically, the input, hidden and output dimensions of the GCN model are 128,64,64 respectively. The input, hidden and output dimensions of the projection layer are 64,64,1 respectively. The ReLU activation function is used in the model. In the output layer of actor network, the Sigmoid() activation function is adopted. And in the output layer of critic network, no activation function is used.

1.2 A.2 MLP-NF model

The MLP-NF model is a two-layer perceptron, of which the input, hidden and output dimensions are 128,64,1 respectively. The ReLU activation function is used in the model. In the output layer of actor network, the Sigmoid() activation function is adopted. On the contrary, no activation function is used in the critic network’s output layer.

1.3 A.3 GCL model

In the graph contrastive learning, one GCL model is constructed. And the model are composed by a two-layer GIN graph neural network and a two-layer perceptron. In GIN module, the input, hidden and output dimensions are 64,32,32 respectively. In perceptron module, the input, hidden and output dimensions are 64,64,64 respectively. The ReLU activation function is used in the model. The representations from this model vill be sent to the graph classifier.

In addition, in order to carry out the GCL learning, the graph level graph augmentations are adopted. Specifically, the data augmentations can be written as [47]:

• $aug1 \!=\! A.RandomChoice([A.RWSampling(num\_seeds$$=\!1000, walk\_length\!=\!10), A.NodeDropping(pn=$$0.1), A.FeatureMasking(pf\!=\!0.1), A.EdgeRemoving$$(pe=0.1),],1)$;

• $aug2 \!=\! A.RandomChoice([A.RWSampling(num\_seeds$$=1000, walk\_length\!=\!10), A.NodeDropping(pn=$$0.1), A.FeatureMasking(pf\!=\!0.1), A.EdgeRemoving$$(pe=0.1)],1)$,

where the PyGCL library is used to finish the graph augmentations.

1.4 A.4 GCN-F model

This model, named GCN sampler, is used to learn to solve MDP (metric dimension problem of graphs) on a set of graphs. The input of the GCN sampler is the node representations generated by GCL model. This sampler’s architecture is the same as GCN-NF model.

1.5 A.5 MLP-F model

This model, named MLP sampler, is used to learn to solve MDP (metric dimension problem of graphs) on a set of graphs. The input of the MLP sampler is the node representations generated by GCL model. This sampler’s architecture is the same as MLP-NF model.

Appendix B: training and evaluation setup

The environment for running is: Intel(R) Core(TM) i7-8750H CPU @ 2.20GHz RAM 16 GB. The PyGCL library, Gurobi, Networkx, networkit and Pytorch software are used.

1.1 B.1 GCL training and evaluation

In the GCL training, Adam optimization algorithm is used, where the learning rate is 0.01, and the DualBranchContrast contrastive model with InfoNCE loss function is adopted [47]. In order to evaluate the quality of GCL learning, the node representations are sent to a graph classifier, where a graph pre-task of graph classification is given. In this paper, the performance of GCL learning is evaluated on test data by SVMEvaluator as a classifier in PyGCL library [47].

1.2 B.2 LS training and evaluation

In LS training process, the sampler is trained to learn to solve MDP on a set of graphs, named train data, and then it is evaluated on test data. Note that, we use PPO reinforcement learning, an unsupervised learning method, to train the sampler. Furthermore, we train the sampler in a manner of mini-batch with batch size 32.

For training of GCN-NF and MLP-NF samplers, the feature for each of the nodes in a graph is ${\textbf {1}}_{1\times d}$ vector with all of the entries are 1. On contrary, for training of GCN-F and MLP-F samplers, the node features in a graph are generated by the well-trained GCL model, named GNN encoder with its frozen parameters.

The Adam optimization algorithm is used to train the samplers, where the learning rate for actor and critic network is 0.01. The number of line search for samplers is 1. The parameter $\lambda$ for computing the advantage function is selected in $\{0.001,0.1,0.2,0.3,0.5\}$. The parameter for clip function is selected in $\{0.1,0.2\}$. We run the training process 1000 episodes on each train data, with 50 as the early stop threshold value. In the reward function, parameters $\alpha _1=2$, $\alpha _2=2$, $\beta =3$. The discount factor for calculating the expected reward is 0.98.

In the process of evaluation, we use the sampler to solve MDP on all of the train data and test data. Then we report the performance of sampler on train data and test data respectively. Where the relative ratio on each of the graphs is computed. Then the average relative ratio is calculated on all of the datasets. Particularly, the number of iteration for add-repair policy and reduce-repair policy is 10.