Tight Bounds for Single-Pass Streaming Complexity of the Set Cover   Problem
  We resolve the space complexity of single-pass streaming algorithms for approximating the classic set cover problem. For finding an $\alpha$-approximate set cover (for any $\alpha= o(\sqrt{n})$) using a single-pass streaming algorithm, we show that $\Theta(mn/\alpha)$ space is both sufficient and necessary (up to an $O(\log{n})$ factor); here $m$ denotes number of the sets and $n$ denotes size of the universe. This provides a strong negative answer to the open question posed by Indyk et al. (2015) regarding the possibility of having a single-pass algorithm with a small approximation factor that uses sub-linear space.   We further study the problem of estimating the size of a minimum set cover (as opposed to finding the actual sets), and establish that an additional factor of $\alpha$ saving in the space is achievable in this case and that this is the best possible. In other words, we show that $\Theta(mn/\alpha^2)$ space is both sufficient and necessary (up to logarithmic factors) for estimating the size of a minimum set cover to within a factor of $\alpha$. Our algorithm in fact works for the more general problem of estimating the optimal value of a covering integer program. On the other hand, our lower bound holds even for set cover instances where the sets are presented in a random order. 
Sepehr Assadi, Sanjeev Khanna, Yang Li
cs.DS