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Better Streaming Algorithms for the Maximum Coverage Problem

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Abstract

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 \(\{1,\ldots ,n \}\), 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:

  • Two \((1-1/e-\epsilon )\) approximation algorithms: One uses \(O(\epsilon ^{-1})\) passes and \(\tilde {O}(\epsilon ^{-2} k)\) space whereas the other uses only a single pass but \(\tilde {O}(\epsilon ^{-2} m)\) space. \(\tilde {O}(\cdot )\) suppresses polylog factors.

  • We show that any approximation factor better than \((1-(1-1/k)^{k})\approx 1-1/e\) in constant passes requires \({\Omega }(m)\) space for constant k even if the algorithm is allowed unbounded processing time. We also demonstrate a single-pass, \((1-\epsilon )\) approximation algorithm using \(\tilde {O}\left (\epsilon ^{-2} m \cdot \min (k,\epsilon ^{-1})\right )\) 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 \({\Omega }(N)\) space for constant k whereas \(\tilde {O}(\epsilon ^{-2}N)\) space is sufficient for a \((1-\epsilon )\) approximation and arbitrary k in a single pass.

  • For regular graphs, we show that \(\tilde {O}(\epsilon ^{-3}k)\) space is sufficient for a \((1-\epsilon )\) approximation in a single pass. We generalize this to a \((\kappa -\epsilon )\) approximation when the ratio between the minimum and maximum degree is bounded below by \(\kappa \).

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Notes

  1. Note that their work addressed the more general problem of maximizing sub-modular functions.

  2. We consider \(1-1/poly(m)\) or \(1-1/poly(n)\) as high probability.

  3. The number of guesses can be reduced to \(\lceil \log _{2} k \rceil \) if the size of the largest set is known since this gives a k approximation of OPT. The size of the large set can be computed in one additional pass if necessary.

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Acknowledgements

We thank Sagar Kale for discussions of related work.

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  1. University of Massachusetts, Amherst, MA, USA

    Andrew McGregor & Hoa T. Vu

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Correspondence to Hoa T. Vu.

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This article is part of the Topical Collection on Special Issue on Database Theory

This work was supported by NSF Awards CCF-0953754, IIS-1251110, CCF-1320719, and a Google Research Award.

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McGregor, A., Vu, H.T. Better Streaming Algorithms for the Maximum Coverage Problem. Theory Comput Syst 63, 1595–1619 (2019). https://doi.org/10.1007/s00224-018-9878-x

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