Towards a Generic Representation of Combinatorial Problems for Learning-Based Approaches

TL;DR

A graph neural network architecture capable of efficiently learning from a fully generic representation of combinatorial problems for learning-based approaches is introduced and Experimental results on four combinatorial problems demonstrate that the architecture achieves performance comparable to dedicated architectures while maintaining generality.

Abstract

In recent years, there has been a growing interest in using learning-based approaches for solving combinatorial problems, either in an end-to-end manner or in conjunction with traditional optimization algorithms. In both scenarios, the challenge lies in encoding the targeted combinatorial problems into a structure compatible with the learning algorithm. Many existing works have proposed problem-specific representations, often in the form of a graph, to leverage the advantages of \textit{graph neural networks}. However, these approaches lack generality, as the representation cannot be easily transferred from one combinatorial problem to another one. While some attempts have been made to bridge this gap, they still offer a partial generality only. In response to this challenge, this paper advocates for progress toward a fully generic representation of combinatorial problems for learning-based approaches. The approach we propose involves constructing a graph by breaking down any constraint of a combinatorial problem into an abstract syntax tree and expressing relationships (e.g., a variable involved in a constraint) through the edges. Furthermore, we introduce a graph neural network architecture capable of efficiently learning from this representation. The tool provided operates on combinatorial problems expressed in the XCSP3 format, handling all the constraints available in the 2023 mini-track competition. Experimental results on four combinatorial problems demonstrate that our architecture achieves performance comparable to dedicated architectures while maintaining generality. Our code and trained models are publicly available at \url{https://github.com/corail-research/learning-generic-csp}.

Figures (2)

• Figure 1: Encoding of a combinatorial problem instance presented as a running example. There are $3$value-vertices depicted (in green) and $2$variable-vertices (in red). As $x_1$ contains values $1$ and $2$ in its domain, they are connected with an edge, and similarly for the domain of $x_2$. There are $2$constraint-vertices (in blue), one for the inequality ($\leq$) and one for the table constraint ($\textsf{ext}$). The figure's gray area illustrates the constraint $3x_1 \leq 4x_2$, highlighted by operators (in orange) and showing a multiplication ($\times$, with feature $3$) of $x_1$ on the right-side ($\textsf{rhs}$) and another ($\times$, with feature $4$) of $x_2$ on the left-side ($\textsf{lhs}$). The $\textsf{rhs}$ and $\textsf{lhs}$ operators clarify equation sides, essential for distinguishing between $3x_1 \leq 4x_2$ and $3x_1 \geq 4x_2$, and link to the associated constraint (e.g., inequality $\leq$). The constraint $\textsc{table}([x_1,x_2],[(1,2),(2,3)])$ is expressed in a similar way. It involves two tuples $t_1$ and $t_2$. Finally, the model-vertex (in yellow) is connected to the two constraints, and to variable $x_1$, as it is part of the objective function.
• Figure 2: Analysis of the generalization ability on larger instances.