The Steiner cycle polytope

This article introduces a new problem called Steiner cycle problem (SCP), closely related to the widely-studied travelling salesman problem (TSP) and the Steiner tree problem. Given an undirected graph G with a penalty associated to each node and a cost to each edge, and given a subset of nodes W, the problem consists of finding a simple cycle in G visiting at least the nodes in W and minimizing the sum of the costs of edges in the cycle plus the penalties of nodes not in the cycle. This problem finds applications in the optimal design of telecommunication systems, and generalizes the known TSP and the weighted Girth problem. We analyze the polyhedral structure associated to the SCP introducing two lifting procedures to extend facet-defining inequalities from the travelling salesman polytope. The obtained results can be applied to the weighted Girth problem. Moreover, they can also be used in a cutting-plane approach to solve the SCP.