An Optimisation Algorithm for Maximum Independent Set with Applications in Map Labelling

We consider the following map labelling problem: given distinct points p1, p2, ..., pn in the plane, find a set of pairwise disjoint axis-parallel squares Q1,Q2, ..., Qn where pi is a corner of Qi. This problem reduces to that of finding a maximum independent set in a graph.We present a branch and cut algorithm for finding maximum independent sets and apply it to independent set instances arising from map labelling. The algorithm uses a new technique for setting variables in the branch and bound tree that implicitly exploits the Euclidean nature of the independent set problems arising from map labelling. Computational experiments show that this technique contributes to controlling the size of the branch and bound tree. We also present a novel variant of the algorithm for generating violated odd-hole inequalities. Using our algorithm we can find provably optimal solutions for map labelling instances with up to 950 cities within modest computing time, a considerable improvement over the results reported on in the literature.