Representative Sets of Product Families

A subfamily \({\cal F}'\) of a set family \({\cal F}\) is said to q-represent \({\cal F}\) if for every \(A \in{\cal F}\) and B of size q such that A ∩ B = ∅ there exists a set \(A' \in{\cal F}'\) such that A′ ∩ B = ∅. In a recent paper [SODA 2014] three of the authors gave an algorithm that given as input a family \({\cal F}\) of sets of size p together with an integer q, efficiently computes a q-representative family \({\cal F'}\) of \({\cal F}\) of size approximately \({p+q \choose p}\), and demonstrated several applications of this algorithm. In this paper, we consider the efficient computation of q-representative sets for product families \({\cal F}\). A family \({\cal F}\) is a product family if there exist families \({\cal A}\) and \({\cal B}\) such that \({\cal F} = \{A \cup B~:~A \in{\cal A}, B \in{\cal B}, A \cap B = \emptyset\}\). Our main technical contribution is an algorithm which given \({\cal A}\), \({\cal B}\) and q computes a q-representative family \({\cal F}'\) of \({\cal F}\). The running time of our algorithm is sublinear in \(|{\cal F}|\) for many choices of \({\cal A}\), \({\cal B}\) 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 \({\cal F}\) 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 \({\cal F}\) from \({\cal A}\) and \({\cal B}\), and then computes the q-representative family \({\cal F}'\) from \({\cal F}\).