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 α-approximate set cover (for α= o(√n)) via a single-pass streaming algorithm, we show that Θ(mn/α) space is both sufficient and necessary (up to an O(logn) 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 (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 α saving in the space is achievable in this case and that this is the best possible. In other words, we show that Θ(mn/α2) space is both sufficient and necessary (up to logarithmic factors) for estimating the size of a minimum set cover to within a factor of α. 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.