Maximum Unique Coverage on Streams: Improved FPT Approximation Scheme and Tighter Space Lower Bound

TL;DR

This paper presents a new fixed-parameter tractable approximation scheme for the Max Unique Coverage problem in data streams, achieving better kernel size and space bounds, and establishes tighter lower bounds on space complexity for certain approximation ratios.

Contribution

It introduces an improved FPT approximation scheme with smaller kernels and space usage, and provides tighter lower bounds on space requirements for streaming algorithms.

Findings

Achieves a $(1- heta)$-approximation with kernel size $ ilde{O}(k r/ heta)$

Uses $ ilde{O}(k^2 r / heta^3)$ space in data streams

Proves $ ilde{ ext{Omega}}(m / k^2)$ space lower bound for certain approximation ratios

Abstract

We consider the Max Unique Coverage problem, including applications to the data stream model. The input is a universe of $n$ elements, a collection of $m$ subsets of this universe, and a cardinality constraint, $k$. The goal is to select a subcollection of at most $k$ sets that maximizes unique coverage, i.e, the number of elements contained in exactly one of the selected sets. The Max Unique Coverage problem has applications in wireless networks, radio broadcast, and envy-free pricing. Our first main result is a fixed-parameter tractable approximation scheme (FPT-AS) for Max Unique Coverage, parameterized by $k$ and the maximum element frequency, $r$, which can be implemented on a data stream. Our FPT-AS finds a $(1-\epsilon)$-approximation while maintaining a kernel of size $\tilde{O}(k r/\epsilon)$, which can be combined with subsampling to use $\tilde{O}(k^2 r / \epsilon^3)$ space overall. This significantly improves on the previous-best FPT-AS with the same approximation, but a kernel of size $\tilde{O}(k^2 r / \epsilon^2)$. In order to achieve our result, we show upper bounds on the ratio of a collection's coverage to the unique coverage of a maximizing subcollection; this is by constructing explicit algorithms that find a subcollection with unique coverage at least a logarithmic ratio of the collection's coverage. We complement our algorithms with our second main result, showing that $\Omega(m / k^2)$ space is necessary to achieve a $(1.5 + o(1))/(\ln k - 1)$-approximation in the data stream. This dramatically improves the previous-best lower bound showing that $\Omega(m / k^2)$ is necessary to achieve better than a $e^{-1+1/k}$-approximation.