A Graph-based Approach for Representing, Comparing, and Classifying Integer Programs

Abstract

It has long been observed that model structure may play an important role in determining the effectiveness of certain solution techniques to mixed integer linear programs (MILPs). One implication is that solution techniques that work well for one class of MILP are often found to be effective on other “similar” problems. Despite the potential benefits, there is limited research in the area of recognizing similarity among MILPs. This paper seeks to address this gap by presenting a framework for classifying and comparing instances of MILPs, based on their mathematical structure. A new graph-based representation of MILPs is proposed, where decision variables and constraints are described by nodes, and arcs denote the presence of decision variables in certain constraints. Inspired by recent advances in the area of deep learning for graph-structured data, two methods of leveraging the MILP graph representations for classification and comparison are provided. In the first relational approach, a graph convolutional network (GCN) is employed to classify these so-called MILP graphs as having come from one of a known number of problem types. The second approach makes use of latent features learned by the GCN to compare graphs to one another directly. As part of the latter approach, we introduce a formal measure of graph-based structural similarity. Empirical studies indicate that both the classification and comparison procedures perform well on a test set. Additional properties of MILP graphs are also explored through a series of case studies.