Quantitative Indicators for Strength of Inequalities with Respect to a Polyhedron, Part II: Applications and Computational Evidence

The first paper explored two strength indicators (extreme point ratio (EPR) and centroid distance (CD)), both of which predict that subtours of the spanning tree in hypergraph polytope STHGP having large cardinality are significantly stronger than the corresponding subtours of small cardinality. In this second paper, we exploit this previously unknown property algorithmically within GeoSteiner, presenting strong computational evidence that the EPR and CD indicators are highly predictive of actual computational strength, at least for subtour inequalities of STHGP. The GeoSteiner package for geometric Steiner trees in the plane uses STHGP to solve the Full Steiner Tree (FST) concatenation problem. The separation algorithms in previous versions of GeoSteiner find violated subtours of only relatively small cardinality, and we examine the underlying algorithmic causes. We present improvements in GeoSteiner specifically designed to strengthen violated subtours by augmentation (instead of reduction), yielding violated subtours of large cardinality. Across all distance metrics and instance classes studied, the computational results are remarkable -- culminating with an optimal solution of a 1,000,000 terminal random Euclidean instance. The conclusion is that the EPR and CD strength indicators presented in the first paper have strong predictive power regarding actual computational strength (at least regarding STHGP subtours). The ability to accurately measure the strength of inequalities has numerous applications of great importance, both in theory and practice.