On constraint sampling in the linear programming approach to approximate linear programming

In the linear programming approach to approximate dynamic programming, one tries to solve a certain linear program - the ALP -, which has a relatively small number K of variables but an intractable number M of constraints. In this paper, we study a scheme that samples and imposes a subset of m ≪ M constraints. A natural question that arises in this context is: How must m scale with respect to K and M in order to ensure that the resulting approximation is almost as good as one given by exact solution of the ALP? We show that, under certain idealized conditions, m can be chosen independently of M and need grow only as a polynomial in K.