Representative Families of Product Families

Abstract

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 3.8408kn^O(1) deterministic algorithm for the Multilinear Monomial Detection (k-MlD) problem. The second is a significant improvement of deterministic dynamic programming algorithms for “connectivity problems” on graphs of bounded treewidth.