Incidence Geometries and the Pass Complexity of Semi-Streaming Set Cover

Set cover, over a universe of size $n$, may be modelled as a data-streaming problem, where the $m$ sets that comprise the instance are to be read one by one. A semi-streaming algorithm is allowed only $O(n\, \mathrm{poly}\{\log n, \log m\})$ space to process this stream. For each $p \ge 1$, we give a very simple deterministic algorithm that makes $p$ passes over the input stream and returns an appropriately certified $(p+1)n^{1/(p+1)}$-approximation to the optimum set cover. More importantly, we proceed to show that this approximation factor is essentially tight, by showing that a factor better than $0.99\,n^{1/(p+1)}/(p+1)^2$ is unachievable for a $p$-pass semi-streaming algorithm, even allowing randomisation. In particular, this implies that achieving a $\Theta(\log n)$-approximation requires $\Omega(\log n/\log\log n)$ passes, which is tight up to the $\log\log n$ factor. These results extend to a relaxation of the set cover problem where we are allowed to leave an $\varepsilon$ fraction of the universe uncovered: the tight bounds on the best approximation factor achievable in $p$ passes turn out to be $\Theta_p(\min\{n^{1/(p+1)}, \varepsilon^{-1/p}\})$. Our lower bounds are based on a construction of a family of high-rank incidence geometries, which may be thought of as vast generalisations of affine planes. This construction, based on algebraic techniques, appears flexible enough to find other applications and is therefore interesting in its own right.