  Semi-algebraic set is a subset of the real space defined by polynomial
equations and inequalities. In this paper, we consider the problem of deciding
whether two given points in a semi-algebraic set are connected. We restrict to
the case when all equations and inequalities are invariant under the action of
the symmetric group and their degrees at most $d<n$, where $n$ is the number of
variables. Additionally, we assume that the two points are in the same
fundamental domain of the action of the symmetric group, by assuming that the
coordinates of two given points are sorted in non-decreasing order. We
construct and analyze an algorithm that solves this problem, by taking
advantage of the group action, and has a complexity being polynomial in $n$.
