2026-07-22T08:26:34Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:7062 2024-03-06T10:39:27Z ddc:004 open_access

Better Streaming Algorithms for the Maximum Coverage Problem McGregor, Andrew Vu, Hoa T. algorithms data streams approximation maximum coverage We study the classic NP-Hard problem of finding the maximum k-set 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 1-1/e in polynomial time. In the streaming-set 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 1-1/e, that use sublinear space o(mn). Our main results are: 1) Two (1-1/e-epsilon) 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-(1-1/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 single-pass, (1-epsilon) approximation algorithm using O(m/epsilon^2 min(k,1/epsilon) polylog(m,n)) space. We also study the maximum k-vertex 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 (1-epsilon) 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 (1-epsilon) approximation in a single pass. We generalize this to a K-epsilon approximation when the ratio between the minimum and maximum degree is bounded below by K. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Andrew McGregor and Hoa T. Vu 2017 Is Part Of LIPIcs, Volume 68, 20th International Conference on Database Theory (ICDT 2017) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICDT.2017.22 urn:nbn:de:0030-drops-70620 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2017.22 eng https://creativecommons.org/licenses/by/3.0/legalcode