  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 $\text{Conf}(K_m,2)$, the configuration space of ordered pairs of
non-colliding particles on the complete graph $K_m$ on $m$ vertices. Together
with the use of Forman's discrete Morse theory, this allows us to describe in
full the cohomology $R$-algebra $H^*(\text{Conf}(K_m,2);R)$ for any commutative
unital ring $R$. As an application we prove that, although $\text{Conf}(K_m,2)$
is outside the "stable" regime, all its topological complexities are maximal
possible when $m\geq4$.
