Learning fine-grained search space pruning and heuristics for combinatorial optimization

Combinatorial optimization problems arise naturally in a wide range of applications from diverse domains. Many of these problems are NP-hard and designing efficient heuristics for them requires considerable time, effort and experimentation. On the other hand, the number of optimization problems in the industry continues to grow. In recent years, machine learning techniques have been explored to address this gap. In this paper, we propose a novel framework for leveraging machine learning techniques to scale-up exact combinatorial optimization algorithms. In contrast to the existing approaches based on deep-learning, reinforcement learning and restricted Boltzmann machines that attempt to directly learn the output of the optimization problem from its input (with limited success), our framework learns the relatively simpler task of pruning the elements in order to reduce the size of the problem instances. In addition, our framework uses only interpretable learning models based on intuitive local features and thus the learning process provides deeper insights into the optimization problem and the instance class, that can be used for designing better heuristics. For the classical maximum clique enumeration problem, we show that our framework can prune a large fraction of the input graph (around 99% of nodes in case of sparse graphs) and still detect almost all of the maximum cliques. Overall, this results in several fold speedups of state-of-the-art algorithms. Furthermore, the classification model used in our framework highlights that the chi-squared value of neighborhood degree has a statistically significant correlation with the presence of a node in a maximum clique, particularly in dense graphs which constitute a significant challenge for modern solvers. We leverage this insight to design a novel heuristic we call ALTHEA for the maximum clique detection problem, outperforming the state-of-the-art for dense graphs.