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 and m sets, our algorithm/protocol achieves a space bound of O(m · n δ log 2 n log m) using O(4 1/δ ) passes/rounds while achieving an approximation factor of O(4 1/δ log n) in polynomial time (for δ = Ω(1/ log n)). If we allow the algorithm/protocol to spend exponential time per pass/round, we achieve an approximation factor of O(4 1/δ ). 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.
Demaine et al. REF studied the trade-off between the number of passes, the approximation ratio, and the space requirement of general streaming algorithms (i.e., not necessarily semi-streaming) for the set cover problem and developed an algorithm that for any δ = Ω(1/ log n), makes O(4 1/δ ) passes over the stream and achieves an O(4 1/δ ρ)approximation using O(mn δ ) space; here ρ is the approximation factor of the off-line algorithm for solving the set cover problem.
On Streaming and Communication Complexity of the Set Cover Problem
DISC
Mathematics Computer Science