Incidence Geometries and the Pass Complexity of Semi-Streaming Set Cover

Set cover, over a universe of size n, may be modelled as a data-streaming problem, where the m sets that comprise the instance are to be read one by one. A semi-streaming algorithm is allowed only O(n poly{log n, log m}) space to process this stream. For each p 1, we give a very simple deterministic algorithm that makes p passes over the input stream and returns an appropriately certified (p + 1)n1/(p+1)-approximation to the optimum set cover. More importantly, we proceed to show that this approximation factor is essentially tight, by showing that a factor better than 0.99 n1/(p+1)/(p + 1)2 is unachievable for a p-pass semi-streaming algorithm, even allowing randomisation. In particular, this implies that achieving a Θ(log n)approximation requires Ω(log n/ log log n) passes, which is tight up to the log log n factor. These results extend to a relaxation of the set cover problem where we are allowed to leave an ε fraction of the universe uncovered: the tight bounds on the best app...

• Type: Journal
• Year: 2016
• Where: SODA
• Authors: Amit Chakrabarti, Anthony Wirth
• Keywords: Algorithms, SODA 2016