Node based compact formulations for the Hamiltonian p-median problem

In this paper, we introduce, study and analyze several classes of compact formulations for the symmetric Hamiltonian p$$ p $$-median problem (Hp$$ p $$MP). Given a positive integer p$$ p $$ and a weighted complete undirected graph G=(V,E)$$ G=\left(V,E\right) $$ with weights on the edges, the Hp$$ p $$MP on G$$ G $$ is to find a minimum weight set of p$$ p $$ elementary cycles partitioning the vertices of G$$ G $$. The advantage of developing compact formulations is that they can be readily used in combination with off-the-shelf optimization software, unlike other types of formulations possibly involving the use of exponentially sized sets of variables or constraints. The main part of the paper focuses on compact formulations for eliminating solutions with less than p$$ p $$ cycles. Such formulations are less well known and studied than formulations which prevent solutions with more than p$$ p $$ cycles. The proposed formulations are based on a common motivation, that is, the formulations contain variables that assign labels to nodes, and prevent less than p$$ p $$ cycles by stating that different depots must have different labels and that nodes in the same cycle must have the same label. We introduce and study aggregated formulations (which consider integer variables that represent the label of the node) and disaggregated formulations (which consider binary variables that assign each node to a given label). The aggregated models are new. The disaggregated formulations are not, although in all of them new enhancements have been included to make them more competitive with the aggregated models. The two main conclusions of this study are: (i) in the context of compact formulations, it is worth looking at the models with integer node variables, which have a smaller size. Despite their weaker LP relaxation bounds, the fewer variables and constraints lead to faster integer resolution, especially when solving instances with more than 50 nodes; (ii) the best of our compact models exhibit a performance that, overall, is comparable to that of the best methods known for the Hp$$ p $$MP (including branch-and-cut algorithms), solving to optimality instances with up to 226 nodes within 1 h. This corroborates our message that the knowledge of the inequalities for preventing less than p$$ p $$ cycles is much less well understood.