Set Cover in the One-pass Edge-arrival Streaming Model

We study the Set Cover problem in the one-pass edge-arrival streaming model. In this model, the input stream consists of a sequence of tuples (S, u), indicating that element u is contained in set S. This setting captures the streaming Dominating Set problem and is more general and harder to solve than the Set Cover set-arrival setting, where entire sets with all their elements arrive in the stream one-by-one. We prove the following results (n is the size of the universe, m is the number of sets): (1) A work by [Khanna, Konrad, ITCS'22] on streaming Dominating Set implies a one-pass (O) over tilde (root n)-approximation algorithm with space (O) over tilde (m) for edge-arrival Set Cover in adversarially ordered streams. We show that this space bound is best possible up to poly-log factors in that every alpha-approximation algorithm, for alpha = Omega(root n), requires space (Omega) over tilde (mn(2)/alpha(4)) in adversarially ordered streams, even if the algorithm is only required to output an alpha-approximation of the size of an optimal cover. (2) As our main result, we give a one-pass (O) over tilde (root n)-approximation algorithm with space (O) over tilde (m/root n) for edge-arrival Set Cover in random order streams. This result together with the lower bound mentioned above establishes a strong separation between the adversarial and random order settings. (3) Finally, in adversarial order streams, we showthat non-trivial algorithms with space o(m) can be achieved at the expense of increased approximation factors (O) over tilde (root n), which is in contrast to the set-arrival setting, where space (O) over tilde (n) is enough for a Theta(root n)-approximation, and space Omega(n) is needed for an o(n/log n)-approximation. We give an alpha-approximation algorithm for one-pass edge-arrival Set Cover with space (O) over tilde (mn/alpha(2)), for every alpha = (Omega) over tilde(root n).