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(root n/ log 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 the number of sets and n denotes the size of the universe. This provides a strong negative answer to the open question posed by Har-Peled et al. [Towards tight bounds for the streaming set cover problem, in Proceedings of the 35th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems (PODS '16), pp. 371-383] regarding the possibility of having a single-pass algorithm with a small approximation factor that uses sublinear 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 savings in the space is achievable in this case and 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.
