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DOI: 10.4230/LIPIcs.ICALP.2019.53
URN: urn:nbn:de:0030-drops-106290
URL: http://drops.dagstuhl.de/opus/volltexte/2019/10629/
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Ene, Alina ; Nguyen, Huy L.

A Nearly-Linear Time Algorithm for Submodular Maximization with a Knapsack Constraint

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LIPIcs-ICALP-2019-53.pdf (0.5 MB)


Abstract

We consider the problem of maximizing a monotone submodular function subject to a knapsack constraint. Our main contribution is an algorithm that achieves a nearly-optimal, 1 - 1/e - epsilon approximation, using (1/epsilon)^{O(1/epsilon^4)} n log^2{n} function evaluations and arithmetic operations. Our algorithm is impractical but theoretically interesting, since it overcomes a fundamental running time bottleneck of the multilinear extension relaxation framework. This is the main approach for obtaining nearly-optimal approximation guarantees for important classes of constraints but it leads to Omega(n^2) running times, since evaluating the multilinear extension is expensive. Our algorithm maintains a fractional solution with only a constant number of entries that are strictly fractional, which allows us to overcome this obstacle.

BibTeX - Entry

@InProceedings{ene_et_al:LIPIcs:2019:10629,
  author =	{Alina Ene and Huy L. Nguyen},
  title =	{{A Nearly-Linear Time Algorithm for Submodular Maximization with a Knapsack Constraint}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{53:1--53:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Christel Baier and Ioannis Chatzigiannakis and Paola Flocchini and Stefano Leonardi},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10629},
  URN =		{urn:nbn:de:0030-drops-106290},
  doi =		{10.4230/LIPIcs.ICALP.2019.53},
  annote =	{Keywords: submodular maximization, knapsack constraint, fast algorithms}
}

Keywords: submodular maximization, knapsack constraint, fast algorithms
Seminar: 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)
Issue Date: 2019
Date of publication: 08.07.2019


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