Neural Maximum Independent Set

Abstract

The emergence of deep learning brought solutions to many difficult problems and has recently motivated new studies that try to solve hard combinatorial optimization problems with machine learning approaches. We propose a framework based on Expert Iteration, an imitation learning method that we apply to solve combinatorial optimization problems on graphs, in particular the Maximum Independent Set problem. Our method relies on training GNNs to recognize how to complete a solution, given a partial solution of the problem as an input. This paper emphasizes some interesting findings such as the introduction of learned nodes features helping the neural network to give relevant solutions. Moreover, we represent the space of good solutions and discuss the ability of GNN’s to solve the problem on a graph without training on it.

The authors acknowledge the support of the ANR as part of the “Investissements d’avenir” program (ANR-19-P3IA-0001, PRAIRIE 3IA Institute) and through the project DELCO (ANR-19-CE23-0016).

Appendices

A Results on other instances

Fig. 3.

The score of the argmax playout each epoch of the learning on each instance. Learning curves with embeddings are highlighted by a plain curve.

Fig. 4.

Distribution of scores of 500 solutions from the training set and an exploration of 500 rollouts with the GNN for each instance, first with only the state, and then with all the features.

B Stochastic exploration method allows to find various good solutions

By giving a set of good solutions, the stochastic exploration method gives us some interesting insights about the distribution of the solutions in a given instance. After several exploration methods, we saved all good solutions of the three best found scores in order to observe the clusters they form (i.e. in the sense of hamming distance between sets). Note that the pool of best solutions for dimacs-frb30-15-1 is the only one that necessitated 5 iterations of labeling/learning phase and the solutions were found in the labeling phase, the stochastic exploration method giving poor solutions as discussed previously.

We sum up the information about the set of solutions we obtained:

• ba200_5: 1400 solutions with score 80, 700 solutions with score 81, 123 solutions with score 82.

• er200_10: 1400 solutions with score 39, 700 solutions with score 40, 94 solutions with score 41.

• dimacs-frb30-15-1: 499 solutions with score 26, 77 solutions with score 27, 4 solutions with score 28.

• bio-SC-LC: 21 solutions with score 966, 5 solutions with score 967, 1 solution with score 968.

In order to observe how the solutions are organized, we first computed all hamming distances between each pair of solution to see how far they are from each other. In Fig. 5, we represented the distribution of all possible hamming distances for the three best found scores for each instance. For er200_10 and dimacs-frb30-15-1, we also represented the graphic in log scale for more readability.

Fig. 5.

The distribution of hamming distances between best found solutions for each instance of the dataset. Sometimes we also give the same graphic in log scale for more readability.

For ba200_5, the solutions are well organized along a Gaussian curve. For er200_10, we observe almost the same phenomenon except that the best solutions (with score 41) are split into three clusters: one big cluster of close solutions, one sparse cluster of partially close solutions, and a small sparse cluster of solutions. For dimacs-frb30-15-1, all best found solutions look very sparse and have very few nodes in common. For bio-SC-LC, the solutions seem to be well organized and not so far from each other. Note that there is no hamming distance for the score 968 since we only obtained one solution for this score.

For each instance, we embedded our pool of solutions into a three dimensional space with a t-SNE [23] in Fig. 6, that allows to observe our previous remarks on the distribution of hamming distances. We can notice that the scatterplots are always nested around the best solutions.

Fig. 6.

The projection of the set of best found solutions for each instance with the t-SNE method. (Color figure online)

Note that for the instances ba200_5 and er200_10, we had to sample the second and third best found solutions (in green and orange) in order to highlight the pool of best solutions (red).

With the instance dimacs-frb30-15-1, we observe that all best found solutions look very sparse in the sense that the solutions have very few nodes in common. This could explain why GNN cannot converge properly into a good solution. The latter remark could explain why our neural network has so much difficulty to learn how to solve the problem on this instance and converge to good solutions.

Since enumerating maximal or maximum independent sets has also been studied in the literature [4, 9, 17], our method provides some interesting tools to obtain good various solutions with the help of a GNN.

C GNNs can export expertise on Max Independent Set from a small graph to a larger graph

The last observation we want to highlight is that once a neural network has learned on a small graph, it performs a little better on other bigger instances it never saw.

In order to do so, we had to remove the footprint features and the embedding features during the learning phase since they had sense only in a particular instance. Thus, the quality of the learning is quite less effective than with all features.

In our experiment, we compare training on ba200_5 and er200_10 and see how much the GNN performs on the other instances (including two other instances: ba100_5 and ba1000_5 constructed in the same way than ba200_5).

After a labeling phase, we make a learning phase and compute the score of the argmax sequence for each other instance using the trained model at each epoch. We then smoothed the curves by computing the average score of the last 5 scores at each epoch, and then represented the approximation ratio between the score of the argmax sequence and the best known result on the instance for each epoch. The resulted curves are represented in Fig. 7.

Fig. 7.

Evolution of approximation ratio (compared to the best known score) on the argmax playout on other graph training on ba200_5 and er200_10.

When training on ba200_5, the quality of the argmax sequence improves quickly at the beginning and then stagnates. When training on er200_10, the qualitywork of the argmax sequence increases slower than with ba200_5 but increases all along the learning. On the other hand, not surprisingly, note that the learning phase did not make any improvement on dimacs-frb30-15-1. Those observations indicate that our GNN was able to transfer some knowledge about Maximum Independent Set on brand new graphs, and is promising for future work.