Representative Sets of Product Families

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 = ∅. 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 ∪ B : A ∈ A, B ∈ B, A ∩ B = ∅}. 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 3.8408n 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. University of Bergen, Norway. {fomin|daniello}@ii.uib.no Institute of Mathematical Sciences, India. {fahad|saket}@imsc.res.in