  The weighted star discrepancy is a quantitative measure for the performance
of point sets in quasi-Monte Carlo algorithms for numerical integration. We
consider polynomial lattice point sets, whose generating vectors can be
obtained by a component-by-component construction to ensure a small weighted
star discre-pancy. Our aim is to significantly reduce the construction cost of
such generating vectors by restricting the size of the set of polynomials from
which we select the components of the vectors. To gain this reduction we
exploit the fact that the weights of the spaces we consider decay very fast.
