  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.
