Quantitative Indicators for Strength of Inequalities with Respect to a Polyhedron, Part I: Theory

Abstract

We study strength of inequalities used in mixed-integer programming, and in branch-and-cut algorithms that solve such problems. Strength is an ethereal property lacking good formal definition, but crucial for computational speed. We review several quantitative indicators proposed in the literature we claim provide a measure of the relative strength of inequalities with respect to a given polyhedron. We evaluate two of these indicators (extreme point ratio (EPR) and centroid distance (CD)) on various facet classes for both the traveling salesman polytope TSP, and spanning tree in hypergraph polytope STHGP, obtaining closed-forms for EPR and CD on each facet class. Within each facet class, the two indicators yield strikingly similar strength rankings, with excellent agreement on which facets are strongest and which are weakest. Both indicators corroborate all known computational experience with both polytopes. The indicators also reveal previously unknown properties of STHGP subtours. We also evaluate EPR and CD for the subtour inequalities of the spanning tree in graphs polytope STGP, obtaining surprising and unexpected results that (at least for STGP and STHGP subtours) lead us to believe EPR to be a more accurate estimate of strength than CD. Applications include: comparing the relative strength of different classes of inequalities; design of rapidly-converging separation algorithms; design or justification for constraint strengthening procedures. The companion paper exploits one of the newly revealed properties of STHGP subtours in GeoSteiner, with detailed computational results. Across all distance metrics and instances studied, these results are remarkable -- culminating with an optimal solution of a 1,000,000 terminal random Euclidean instance. This confirms these indicators to be highly predictive and strongly correlated with actual computational strength.