  In order to deal with the high development time of exact and approximation
algorithms for NP-hard combinatorial optimisation problems and the high running
time of exact solvers, deep learning techniques have been used in recent years
as an end-to-end approach to find solutions. However, there are issues of
representation, generalisation, complex architectures, interpretability of
models for mathematical analysis etc. using deep learning techniques. As a
compromise, machine learning can be used to improve the run time performance of
exact algorithms in a matheuristics framework. In this paper, we use a pruning
heuristic leveraging machine learning as a pre-processing step followed by an
exact Integer Programming approach. We apply this approach to sparsify
instances of the classical travelling salesman problem. Our approach learns
which edges in the underlying graph are unlikely to belong to an optimal
solution and removes them, thus sparsifying the graph and significantly
reducing the number of decision variables. We use carefully selected features
derived from linear programming relaxation, cutting planes exploration,
minimum-weight spanning tree heuristics and various other local and statistical
analysis of the graph. Our learning approach requires very little training data
and is amenable to mathematical analysis. We demonstrate that our approach can
reliably prune a large fraction of the variables in TSP instances from
TSPLIB/MATILDA (>85%$) while preserving most of the optimal tour edges. Our
approach can successfully prune problem instances even if they lie outside the
training distribution, resulting in small optimality gaps between the pruned
and original problems in most cases. Using our learning technique, we discover
novel heuristics for sparsifying TSP instances, that may be of independent
interest for variants of the vehicle routing problem.
