Neural Models for Output-Space Invariance in Combinatorial Problems

TL;DR

This paper introduces novel GNN-based methods to achieve output-space invariance in combinatorial problems, enabling models to generalize across different sizes of output value-sets, such as in sudoku and graph coloring.

Contribution

The authors propose two new GNN architectures that extend existing models to handle varying output value-set sizes, improving generalization in combinatorial problem solving.

Findings

Both models outperform generic neural reasoners on multiple problems.

The binarized model excels with small value-sets, while the multi-valued model is more efficient for larger sets.

Trade-offs exist between performance and memory efficiency depending on the model used.

Abstract

Recently many neural models have been proposed to solve combinatorial puzzles by implicitly learning underlying constraints using their solved instances, such as sudoku or graph coloring (GCP). One drawback of the proposed architectures, which are often based on Graph Neural Networks (GNN), is that they cannot generalize across the size of the output space from which variables are assigned a value, for example, set of colors in a GCP, or board-size in sudoku. We call the output space for the variables as 'value-set'. While many works have demonstrated generalization of GNNs across graph size, there has been no study on how to design a GNN for achieving value-set invariance for problems that come from the same domain. For example, learning to solve 16 x 16 sudoku after being trained on only 9 x 9 sudokus. In this work, we propose novel methods to extend GNN based architectures to achieve value-set invariance. Specifically, our model builds on recently proposed Recurrent Relational Networks. Our first approach exploits the graph-size invariance of GNNs by converting a multi-class node classification problem into a binary node classification problem. Our second approach works directly with multiple classes by adding multiple nodes corresponding to the values in the value-set, and then connecting variable nodes to value nodes depending on the problem initialization. Our experimental evaluation on three different combinatorial problems demonstrates that both our models perform well on our novel problem, compared to a generic neural reasoner. Between two of our models, we observe an inherent trade-off: while the binarized model gives better performance when trained on smaller value-sets, multi-valued model is much more memory efficient, resulting in improved performance when trained on larger value-sets, where binarized model fails to train.