Better Streaming Algorithms for the Maximum Coverage Problem

TL;DR

This paper develops new streaming algorithms for the maximum coverage problem, achieving near-optimal approximation ratios with sublinear space, and extends the results to dynamic graph models with insertions and deletions.

Contribution

It introduces multiple streaming algorithms with near-optimal approximation guarantees and space complexity for maximum coverage and vertex coverage problems, including dynamic graph models.

Findings

Two $(1-1/e- ext{epsilon})$ approximation algorithms with different space and pass complexities.

Single-pass $(1- ext{epsilon})$ approximation algorithms with sublinear space.

Lower bounds showing space requirements for better-than-$(1-(1-1/k)^k)$ approximations.

Abstract

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 $\{1,\ldots,n \}$, 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 $\tilde{O}(\epsilon^{-2} k)$ space whereas the other uses only a single pass but $\tilde{O}(\epsilon^{-2} m)$ space. We show that any approximation factor better than $(1-(1-1/k)^k)$ 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 $\tilde{O}(\epsilon^{-2} m \cdot \min(k,\epsilon^{-1}))$ 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 $\tilde{O}(\epsilon^{-2}N)$ space is sufficient for a $(1-\epsilon)$ approximation and arbitrary $k$ in a single pass. For regular graphs, we show that $\tilde{O}(\epsilon^{-3}k)$ 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 $\kappa$.