Ali Vakilian:Streaming Algorithms for Set Cover Problem

Wednesday, July 20, 2016 - 4:00pm to 5:00pm
Ali Vakilian

We consider the classic Set Cover problem in the data stream model. For $n$ elements and $m$ sets we give a $O(1/\delta)$-pass algorithm with a strongly sub-linear $\tldO(mn^{\delta})$ space and logarithmic approximation factor. 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 $\tldOmega(m n^{\delta})$ space. 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 sub-linear space bound. 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.

This is based on joint work with Erik Demaine, Sariel Har-Peled, Piotr Indyk and Sepideh Mahabadi.