  This paper presents two algorithms. In their simplest form, the first
algorithm decides the existence of a pointed homotopy between given simplicial
maps f, g from X to Y and the second computes the group $[\Sigma X,Y]^*$ of
pointed homotopy classes of maps from a suspension; in both cases, the target Y
is assumed simply connected and the algorithms run in polynomial time when the
dimension of X is fixed. More generally, these algorithms work relative to a
subspace A of X, fibrewise over a simply connected B and also equivariantly
when all spaces are equipped with a free action of a fixed finite group G.
