Column-generation based bounds for the homogeneous areas problem

Given an undirected graph and a collection of vertex subsets with suitable costs, we consider the problem of partitioning the graph into subgraphs of limited cost, splitting as little as possible the given subsets among different subgraphs. This problem originates from the organization of a region (the graph) including several towns (the vertices) into administrative areas (the subgraphs). The officers assigned to each area take care of activities which involve several towns at a time (the subsets). An activity involving towns from more areas engages the officers of all those areas, leading to redundancies which must be minimized. This paper introduces a column generation approach to compute a lower bound for the problem. Since the pricing subproblem is NP-hard, we solve it with a Tabu Search algorithm, before applying a suitably strengthened multi-commodity flow formulation. Moreover, we also compute an upper bound for the overall problem with a primal heuristic based on the idea of diving and limited discrepancy search. The computational results refer to two real-world instances, a class of realistic instances derived from them, and two different classes of random instances.