Better Streaming Algorithms for the Maximum Coverage Problem

We study the classic NP-Hard problem of finding the maximum k-set coverage in the data stream model: given a set system of m sets that are subsets of a universe {1, . . . , n}, find the k sets that cover the most number of distinct elements. The problem can be approximated up to a factor 1 - 1/e in polynomial time. In the streaming-set model, the sets and their elements are revealed online. The main goal of our work is to design algorithms, with approximation guarantees as close as possible to 1 - 1/e, that use sublinear space o(mn). Our main results are: Two (1 - 1/e - epsilon) approximation algorithms: One uses O(epsilon(-1)) passes and (O) over tilde(epsilon(-2)k) space whereas the other uses only a single pass but (O) over tilde(epsilon(-2)m) space. (O) over tilde(center dot) suppresses polylog factors. We show that any approximation factor better than (1 - (1 - 1/k) k) approximate to 1 - 1/e in constant passes requires Omega(m) space for constant k even if the algorithm is allowed unbounded processing time. We also demonstrate a single-pass, (1 - epsilon) approximation algorithm using (O) over tilde (epsilon(-2)m.min(k, epsilon(-1))) space. We also study the maximum k-vertex coverage problem in the dynamic graph stream model. In this model, the stream consists of edge insertions and deletions of a graph on N vertices. The goal is to find k vertices that cover the most number of distinct edges. We show that any constant approximation in constant passes requires Omega(N) space for constant k whereas (O) over tilde(epsilon N-2) space is sufficient for a (1 - epsilon) approximation and arbitrary k in a single pass. For regular graphs, we show that (O) over tilde(epsilon(-3)k) space is sufficient for a (1 - epsilon)approximation in a single pass. We generalize this to a (kappa - epsilon) approximation when the ratio between the minimum and maximum degree is bounded below by kappa.