An exact approach for the multi-constraint graph partitioning problem

In this work, a multi-constraint graph partitioning problem is introduced. The input is an undirected graph with costs on the edges and multiple weights on the nodes. The problem calls for a partition of the node set into a fixed number of clusters, such that each cluster satisfies a collection of node weight constraints, and the total cost of the edges whose end nodes are in the same cluster is minimized. It arises as a sub-problem of an integrated vehicle and pollster problem from a real-world application. Two integer programming formulations are provided, and several families of valid inequalities associated with the respective polyhedra are proved. An exact algorithm based on Branch & Bound and cutting planes is proposed, and it is tested on real-world instances.