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

TL;DR

This work studies Max Unique Coverage, a variant of Max Coverage focusing on elements covered by exactly one of the chosen $k$ sets, with a data-stream perspective. It introduces a kernelization-based fixed-parameter tractable approximation scheme (FPT-AS) parameterized by $k$ and the maximum frequency $r$, achieving a $(1-\varepsilon)$-approximation with a kernel of size $\left\lceil k r (\phi+1)/\varepsilon \right\rceil$ and a streaming-space bound of $\tilde{O}(\phi k^2 r/\varepsilon^3)$, significantly improving previous kernels and enabling practical streaming solutions. A key component is a new bound on the unique coverage ratio $\phi$ via offline polynomial-time algorithms (UniqueGreedy, UniqueGreedyFreq, UniqueGreedySize) that feed into the kernel, yielding tighter guarantees than prior work. The paper also proves a substantial space-lower-bound: any randomized streaming algorithm achieving a $(1.5+o(1))/(\ obreak ln k - 1)$-approximation requires $\Omega(m/k^2)$ space, highlighting a sharp gap that motivates further exploration of parameterized streaming for coverage problems. Together, these results advance the design of space-efficient streaming schemes and sharpen fundamental limits for Max Unique Coverage in data streams.

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 $(1-ε)$-approximation while maintaining a kernel of size $\tilde{O}(k r/ε)$, which can be combined with subsampling to use $\tilde{O}(k^2 r / ε^3)$ space overall. This significantly improves on the previous-best FPT-AS with the same approximation, but a kernel of size $\tilde{O}(k^2 r / ε^2)$. 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 $Ω(m / k^2)$ 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 $Ω(m / k^2)$ is necessary to achieve better than a $e^{-1+1/k}$-approximation.

Theorems & Definitions (9)

• Lemma 1: Subsampling Approach mcgregor2021maximum
• Lemma 2: Kernelization Lemma
• Theorem 7
• Lemma 8
• Theorem 9
• Theorem 10
• Theorem 13
• Theorem 17
• Theorem 18