Abstract
We study the classic NPHard problem of finding the maximum kset coverage in the data stream model: given a set system of m sets that are subsets of a universe {1,...,n}, find the k sets that cover the most number of distinct elements. The problem can be approximated up to a factor 11/e in polynomial time. In the streamingset model, the sets and their elements are revealed online. The main goal of our work is to design algorithms, with approximation guarantees as close as possible to 11/e, that use sublinear space o(mn). Our main results are: 1) Two (11/eepsilon) approximation algorithms: One uses O(1/epsilon) passes and O(k/epsilon^2 polylog(m,n)) space whereas the other uses only a single pass but O(m/epsilon^2 polylog(m,n)) space. 2) We show that any approximation factor better than (1(11/k)^k) in constant passes require space that is linear in m for constant k even if the algorithm is allowed unbounded processing time. We also demonstrate a singlepass, (1epsilon) approximation algorithm using O(m/epsilon^2 min(k,1/epsilon) polylog(m,n)) space.
We also study the maximum kvertex coverage problem in the dynamic graph stream model. In this model, the stream consists of edge insertions and deletions of a graph on N vertices. The goal is to find k vertices that cover the most number of distinct edges. We show that any constant approximation in constant passes requires space that is linear in N for constant k whereas O(N/epsilon^2 polylog(m,n)) space is sufficient for a (1epsilon) approximation and arbitrary k in a single pass. For regular graphs, we show that O(k/epsilon^3 polylog(m,n)) space is sufficient for a (1epsilon) approximation in a single pass. We generalize this to a Kepsilon approximation when the ratio between the minimum and maximum degree is bounded below by K.
BibTeX  Entry
@InProceedings{mcgregor_et_al:LIPIcs:2017:7062,
author = {Andrew McGregor and Hoa T. Vu},
title = {{Better Streaming Algorithms for the Maximum Coverage Problem}},
booktitle = {20th International Conference on Database Theory (ICDT 2017)},
pages = {22:122:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770248},
ISSN = {18688969},
year = {2017},
volume = {68},
editor = {Michael Benedikt and Giorgio Orsi},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/7062},
URN = {urn:nbn:de:0030drops70620},
doi = {10.4230/LIPIcs.ICDT.2017.22},
annote = {Keywords: algorithms, data streams, approximation, maximum coverage}
}
Keywords: 

algorithms, data streams, approximation, maximum coverage 
Collection: 

20th International Conference on Database Theory (ICDT 2017) 
Issue Date: 

2017 
Date of publication: 

17.03.2017 