The computation of integrals in higher dimensions and on general domains, when no explicit cubature rules are known, can be "easily" addressed by means of the quasi-Monte Carlo method. The method, simple in its formulation, becomes computationally inefficient when the space dimension is growing and the integration domain is particularly complex. In this paper we present two new approaches to the quasi-Monte Carlo method for cubature based on  nonnegative least squares and  approximate Fekete points. The main idea is to use less points and especially  good points for solving the system of the moments.  Good points are here intended as points with good interpolation properties, due to the strict connection between interpolation and cubature. Numerical experiments show that, in average, just a tenth of the points should be used to maintain the same approximation order of the quasi-Monte Carlo method. The method has been satisfactorily applied to 2- and 3-dimensional problems on quite complex domains.