We introduce an algorithm that constructs a discrete gradient field on any
simplicial complex. We show that, in all situations, the gradient field is
maximal possible and, in a number of cases, optimal. We make a thorough
analysis of the resulting gradient field in the case of Munkres' discrete model
for Conf(Km,2), the configuration space of ordered pairs of
non-colliding particles on the complete graph Km on m vertices. Together
with the use of Forman's discrete Morse theory, this allows us to describe in
full the cohomology R-algebra H∗(Conf(Km,2);R) for any commutative
unital ring R. As an application we prove that, although Conf(Km,2)
is outside the "stable" regime, all its topological complexities are maximal
possible when m≥4.Comment: 29 pages, 18 figure