Maximum Coverage in Sublinear Space, Faster

TL;DR

This paper improves the efficiency of a sublinear-space streaming algorithm for maximum coverage, reducing time costs and simplifying mechanisms while maintaining approximation quality and low space complexity.

Contribution

The authors optimize a sublinear-space streaming algorithm for maximum coverage by reducing time costs, refining error bounds, and simplifying the sketching mechanism without sacrificing performance.

Findings

Optimizations do not alter space complexity or approximation quality.

Refined error bounds allow for lower independence requirements.

Simplified F0-sketching maintains solution quality with faster execution.

Abstract

Given a collection of $m$ sets from a universe $\mathcal{U}$, the Maximum Set Coverage problem consists of finding $k$ sets whose union has largest cardinality. This problem is NP-Hard, but the solution can be approximated by a polynomial time algorithm up to a factor $1-1/e$. However, this algorithm does not scale well with the input size. In a streaming context, practical high-quality solutions are found, but with space complexity that scales linearly with respect to the size of the universe $|\mathcal{U}|$. However, one randomized streaming algorithm has been shown to produce a $1-1/e-\varepsilon$ approximation of the optimal solution with a space complexity that scales only poly-logarithmically with respect to $m$ and $|\mathcal{U}|$. In order to achieve such a low space complexity, the authors used a technique called subsampling, based on independent-wise hash functions, and $F_0$-sketching. This article focuses on this sublinear-space algorithm and introduces methods to reduce the time cost of subsampling. Firstly, we give some optimizations that do not alter the space complexity, number of passes and approximation quality of the original algorithm. In particular, we reanalyze the error bounds to show that the original independence factor of $\Omega(\varepsilon^{-2} k \log m)$ can be fine-tuned to $\Omega(k \log m)$. Secondly we show that $F_0$-sketching can be replaced by a much more simple mechanism. Finally, our experimental results show that even a pairwise-independent hash-function sampler does not produce worse solution than the original algorithm, while running significantly faster by several orders of magnitude.