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

Abstract

"Strength" is an important property of inequalities used in mixed-integer optimization, both in theory and practice. Unfortunately, no good formal characterization for strength exists, nor is it well-understood. The first paper explored two strength indicators (extreme point ratio (EPR) and centroid distance (CD)), applying them to the subtour inequalities of the spanning tree in hypergraph polytope STHGP. Although it was known that subtour inequalities of small cardinality were strong, EPR and CD agree that subtour inequalities of large cardinality are significantly stronger. In this second paper, we exploit this previously unknown property algorithmically, presenting strong computational evidence that the EPR and CD indicators are highly predictive of actual computational strength. Previous branch-and-cut implementations for optimizing over STHGP find violated subtour inequalities of only relatively small cardinality, strengthening only by reducing the cardinality of violated subtours. We present new methods that strengthen violated subtour inequalities by augmentation (instead of reduction). Combining strengthening via reduction and augmentation yields violated subtour inequalities of both small and large cardinality, covering both classes deemed "strong" by EPR and CD. Across all instance classes studied, the computational results are remarkable -- culminating with an optimal solution of a 1,000,000 terminal random Euclidean Steiner tree 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.