Representative Sets of Product Families

A subfamily F' of a set family F is said to q-represent F if for every A is an element of F and B of size q such that A boolean AND B = empty set there exists a set A' is an element of F' such that A' boolean AND B = empty set. In a recent paper [SODA 2014] three of the authors gave an algorithm that given as input a 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)), and demonstrated several applications of this algorithm. In this paper, we consider the efficient computation of q-representative sets for product families F. A family F is a product family if there exist families A and B such that F = {A boolean OR B : A is an element of A, B is an element of B, A boolean AND B = empty set}. Our main technical contribution is an algorithm which given A, B and q computes a q-representative family F' of F. The running time of our algorithm is sublinear in vertical bar F vertical bar for many choices of A, B and q which occur naturally in several dynamic programming algorithms. We also give an algorithm for the computation of q-representative sets 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 sets for product families. The first is a 3.8408(k)n(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.