The multiple steiner TSP with cyclic order on terminals: valid inequalities and polyhedra

Abstract This paper deals with a variant of the Traveling Salesman Problem (TSP), called the Multiple Steiner TSP with Order Constraints (MSTSPOC). Consider an undirected graph with nonnegative weights on the edges, and a set of salesmen such that with each salesman is associated a set of ordered terminals. The MSTSPOC consists in finding a minimum-weight subgraph containing for each salesman a tour going in order through its terminals. We study the polytope associated with the Integer Linear Programming (ILP) formulation proposed in Borne et al. (2013). We characterize when the basic inequalities define facets. We also describe new valid inequalities along with necessary conditions and sufficient conditions for these inequalities to be facet-defining. Further families of valid inequalities, coming from closely related problems, are also discussed. The theoretical results presented in this paper are computationally tested in a companion paper (Taktak 2024).