A subfamily F′ of a set family F is said to q - represent F if for every A ∈ F and B of size q such that A ∩ B = ∅ there exists a set A′ ∈ F′ such that A′ ∩ B = ∅. Recently, we provided an algorithm that, for a given family F of sets of size p together with an integer q , efficiently computes a q -representative family F′ of F of size approximately (p+q p). In this article, we consider the efficient computation of q -representative families for product families F . A family F is a product family if there exist families A and B such that F = { A , ∪, B : A ∈ A , B ∈ B , A , ∩, B = ∅}. Our main technical contribution is an algorithm that, given A , B and q , computes a q -representative family F′ of F . The running time of our algorithm is sublinear in | F | for many choices of A , B , and q that occur naturally in several dynamic programming algorithms. We also give an algorithm for the computation of q -representative families for product families F in the more general setting where q -representation also involves independence in a matroid in addition to disjointness. This algorithm considerably outperforms the naive approach where one first computes F from A and B and then computes the q -representative family F′ from F . We give two applications of our new algorithms for computing q -representative families for product families. The first is a 3.8408 k n O (1) deterministic algorithm for the M ultilinear M onomial D etection ( k -M l D) problem. The second is a significant improvement of deterministic dynamic programming algorithms for “connectivity problems” on graphs of bounded treewidth.