Stable set reformulations for the degree preserving spanning tree problem

Abstract

Let $G=(V,E)$ be a connected undirected graph and assume that a spanning tree is available for it. Any vertex in this tree is called degree preserving if it has the same degree in the graph and in the tree. Building upon this concept, the Degree Preserving Spanning Tree Problem (DPSTP) asks for a spanning tree of $G$ with as many degree preserving vertices as possible. DPSTP is very much intertwined with the most important application for it, so far, i.e., the online monitoring of arc flows in a water distribution network, what serves us as an additional motivation for our investigation. We show that degree preserving vertices correspond to a stable set of a properly defined DPSTP conflict graph. This, in turn, allows us to use valid inequalities for the Stable Set Polytope (SSP) and thus strengthen the DPSTP formulations previously suggested in the literature. In doing so, the Branch-and-cut and Benders Decomposition algorithms suggested for these formulations are greatly enhanced. Additionally, as a further contribution, we also introduce a new and very challenging set of DPSTP test instances, given by graphs that closely resemble water distribution networks. For these instances, the new algorithms very clearly outperform all previous DPSTP algorithms. They not only run faster than their competitors but are also less dependent on good initial DPSTP primal bounds. For previous DPSTP test sets, the new algorithms lag behind the best existing algorithm for them, which is based on Lagrangian relaxation. However, as compared to additional DPSTP previous algorithms that do not benefit from SSP inequalities, the new ones come much closer to the Lagrangian relaxation one.