Representative Sets of Product Families

Abstract

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 ${\cal F}$.

We give two applications of our new algorithms for computing q-representative sets for product families. The first is a $3.8408^kn^{{\mathcal{O}}(1)}$ 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.