On Streaming and Communication Complexity of the Set Cover Problem

We develop the first streaming algorithm and the first two-party communication protocol that uses a constant number of passes/rounds and sublinear space/communication for logarithmic approximation to the classic Set Cover problem. Specifically, for n elements andm sets, our algorithm/protocol achieves a space bound of O(m · n log n logm) using O(4) passes/rounds while achieving an approximation factor of O(4 logn) in polynomial time (for δ = Ω(1/ logn)). If we allow the algorithm/protocol to spend exponential time per pass/round, we achieve an approximation factor of O(4). Our approach uses randomization, which we show is necessary: no deterministic constant approximation is possible (even given exponential time) using o(mn) space. These results are some of the first on streaming algorithms and efficient two-party communication protocols for approximation algorithms. Moreover, we show that our algorithm can be applied to multi-party communication model.