# Towards Tight Bounds for the Streaming Set Cover Problem $\renewcommand{\Re}{{\rm I\!\hspace{-0.025em} R}} \newcommand{\tldOmega}{\widetilde{\Omega}}% \newcommand{\tldO}{\widetilde{O}} \newcommand{\SetX}{\mathsf{X}} \newcommand{\eps}{\varepsilon} \newcommand{\VorX}[1]{\mathcal{V} \pth{#1}} \newcommand{\Polygon}{\mathsf{P}} \newcommand{\IntRange}[1]{[ #1 ]} \newcommand{\Space}{\overline{\mathsf{m}}} \newcommand{\pth}[2][\!]{#1\left({#2}\right)}$

Sariel Har-Peled, Piotr Indyk, Sepideh Mahabadi, and Ali Vakilian.
We consider the classic SetCover problem in the data stream model. For $n$ elements and $m$ sets ($m\geq n$) we give a $O(1/\delta)$-pass algorithm with a strongly sub-linear $\tldO(mn^{\delta})$ space and logarithmic approximation factor. This yields a significant improvement over the earlier algorithm of Demaine etal that uses exponentially larger number of passes. We complement this result by showing that the tradeoff between the number of passes and space exhibited by our algorithm is tight, at least when the approximation factor is equal to $1$. Specifically, we show that any algorithm that computes set cover exactly using $({1 \over 2\delta}-1)$ passes must use $\tldOmega(mn^{\delta})$ space in the regime of $m=O(n)$. Furthermore, we consider the problem in the geometric setting where the elements are points in $\Re^2$ and sets are either discs, axis-parallel rectangles, or fat triangles in the plane, and show that our algorithm (with a slight modification) uses the optimal $\tldO(n)$ space to find a logarithmic approximation in $O(1/\delta)$ passes. Finally, we show that any randomized one-pass algorithm that distinguishes between covers of size 2 and 3 must use a linear (i.e., $\Omega(mn)$) amount of space. This is the first result showing that a randomized, approximate algorithm cannot achieve a space bound that is sublinear in the input size. This indicates that using multiple passes might be necessary in order to achieve sub-linear space bounds for this problem while guaranteeing small approximation factors.
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Last modified: Mon Jul 6 13:05:52 CDT 2015