# The Hamiltonian p-median Problem: Polyhedral Results and Branch-and-Cut Algorithm


In this paper we study the Hamiltonian $$p$$-median problem, in which a weighted graph on $$n$$ vertices is to be partitioned into $$p$$ simple cycles of minimum total weight. We introduce two new families of valid inequalities for a formulation of the problem in the space of natural edge variables. Each one of the families forbids solutions to the 2-factor relaxation of the problem that have less than $$p$$ cycles. The inequalities in one of the families are associated with large simple cycles of the underlying graph and generalize known inequalities associated to Hamiltonian cycles. The other family involves inequalities for the case with $$p=\left\lfloor n/3\right\rfloor$$, associated with edge cuts and multi-cuts whose shores have specific cardinalities. We identify inequalities from both families that define facets of the polytope associated with the problem in the edge variables space.
We design branch-and-cut algorithms based on these families of inequalities and on inequalities associated with 2-opt moves removing sub-optimal solutions.
Computational experiments on benchmark instances show that our algorithm exhibits a comparable performance with respect to existing exact methods from the literature; moreover our algorithm solves to optimality new instances with up to 400 vertices.