Maximum Coverage in the Data Stream Model: Parameterized and Generalized

Abstract

We present algorithms for the Max-Cover and Max-Unique-Cover problems in the data stream model. The input to both problems are $m$ subsets of a universe of size $n$ and a value $k\in [m]$. In Max-Cover, the problem is to find a collection of at most $k$ sets such that the number of elements covered by at least one set is maximized. In Max-Unique-Cover, the problem is to find a collection of at most $k$ sets such that the number of elements covered by exactly one set is maximized. Our goal is to design single-pass algorithms that use space that is sublinear in the input size. Our main algorithmic results are: If the sets have size at most $d$, there exist single-pass algorithms using $\tilde{O}(d^{d+1}{k}^d)$ space that solve both problems exactly. This is optimal up to polylogarithmic factors for constant $d$. If each element appears in at most $r$ sets, we present single pass algorithms using $\tilde{O}(k^{2}r/{\epsilon }^3)$ space that return a $1+\epsilon$ approximation in the case of Max-Cover. We also present a single-pass algorithm using slightly more memory, i.e., $\tilde{O}(k^{3}r/{\epsilon }^4)$ space, that $1+\epsilon$ approximates Max-Unique-Cover. In contrast to the above results, when $d$ and $r$ are arbitrary, any constant pass $1+\epsilon$ approximation algorithm for either problem requires $\Omega ({\epsilon }^{-2}m)$ space but a single pass $O({\epsilon }^{-2}mk)$ space algorithm exists. In fact any constant-pass algorithm with an approximation better than $e/(e-1)$ and $e^{1-1/k}$ for Max-Cover and Max-Unique-Cover respectively requires $\Omega (m/k^2)$ space when $d$ and $r$ are unrestricted. En route, we also obtain an algorithm for a parameterized version of the streaming Set-Cover problem.

Conference version to appear at ICDT 2021

References in corpus (1)

• Approximating Semi-Matchings in Streaming and in Two-Party Communication

Cited by in corpus (1)

• Matroid-Constrained Vertex Cover