Representative sets of product families

Abstract

A subfamily of F′ a set family F is said to q-represent F if for every A ε F and B of size q such that A∩B=∅ there exists a set A' ε F' such that A'∩B=∅. 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' F of 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 ∪B : A' ε A,B ε B, A ∩B = emptyv;}. 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 |F|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 3.8408knO(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. © 2014 Springer-Verlag Berlin Heidelberg.