3 Search Results for "Norouzi-Fard, Ashkan"

Document
DOI: 10.4230/LIPIcs.SEA.2023.21
Maximum Coverage in Sublinear Space, Faster

Authors: Stephen Jaud, Anthony Wirth, and Farhana Choudhury

Published in: LIPIcs, Volume 265, 21st International Symposium on Experimental Algorithms (SEA 2023)

Abstract
Given a collection of m sets from a universe 𝒰, the Maximum Set Coverage problem consists of finding k sets whose union has largest cardinality. This problem is NP-Hard, but the solution can be approximated by a polynomial time algorithm up to a factor 1-1/e. However, this algorithm does not scale well with the input size. In a streaming context, practical high-quality solutions are found, but with space complexity that scales linearly with respect to the size of the universe n = |𝒰|. However, one randomized streaming algorithm has been shown to produce a 1-1/e-ε approximation of the optimal solution with a space complexity that scales only poly-logarithmically with respect to m and n. In order to achieve such a low space complexity, the authors used two techniques in their multi-pass approach: - F₀-sketching, allows to determine with great accuracy the number of distinct elements in a set using less space than the set itself. - Subsampling, consists of only solving the problem on a subspace of the universe. It is implemented using γ-independent hash functions. This article focuses on the sublinear-space algorithm and highlights the time cost of these two techniques, especially subsampling. We present optimizations that significantly reduce the time complexity of the algorithm. Firstly, we give some optimizations that do not alter the space complexity, number of passes and approximation quality of the original algorithm. In particular, we reanalyze the error bounds to show that the original independence factor of Ω(ε^{-2} k log m) can be fine-tuned to Ω(k log m); we also show how F₀-sketching can be removed. Secondly, we derive a new lower bound for the probability of producing a 1-1/e-ε approximation using only pairwise independence: 1- (4/(c k log m)) compared to 1-(2e/(m^{ck/6})) with Ω(k log m)-independence. Although the theoretical guarantees are weaker, suggesting the approximation quality would suffer, for large streams, our algorithms perform well in practice. Finally, our experimental results show that even a pairwise-independent hash-function sampler does not produce worse solution than the original algorithm, while running significantly faster by several orders of magnitude.
Cite as

Stephen Jaud, Anthony Wirth, and Farhana Choudhury. Maximum Coverage in Sublinear Space, Faster. In 21st International Symposium on Experimental Algorithms (SEA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 265, pp. 21:1-21:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{jaud_et_al:LIPIcs.SEA.2023.21,
  author =	{Jaud, Stephen and Wirth, Anthony and Choudhury, Farhana},
  title =	{{Maximum Coverage in Sublinear Space, Faster}},
  booktitle =	{21st International Symposium on Experimental Algorithms (SEA 2023)},
  pages =	{21:1--21:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-279-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{265},
  editor =	{Georgiadis, Loukas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2023.21},
  URN =		{urn:nbn:de:0030-drops-183715},
  doi =		{10.4230/LIPIcs.SEA.2023.21},
  annote =	{Keywords: streaming algorithms, subsampling, maximum set cover, k-wise independent hash functions}
}
Document
DOI: 10.4230/LIPIcs.ESA.2022.52
Submodular Maximization Subject to Matroid Intersection on the Fly

Authors: Moran Feldman, Ashkan Norouzi-Fard, Ola Svensson, and Rico Zenklusen

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract
Despite a surge of interest in submodular maximization in the data stream model, there remain significant gaps in our knowledge about what can be achieved in this setting, especially when dealing with multiple constraints. In this work, we nearly close several basic gaps in submodular maximization subject to k matroid constraints in the data stream model. We present a new hardness result showing that super polynomial memory in k is needed to obtain an o(k/(log k))-approximation. This implies near optimality of prior algorithms. For the same setting, we show that one can nevertheless obtain a constant-factor approximation by maintaining a set of elements whose size is independent of the stream size. Finally, for bipartite matching constraints, a well-known special case of matroid intersection, we present a new technique to obtain hardness bounds that are significantly stronger than those obtained with prior approaches. Prior results left it open whether a 2-approximation may exist in this setting, and only a complexity-theoretic hardness of 1.91 was known. We prove an unconditional hardness of 2.69.
Cite as

Moran Feldman, Ashkan Norouzi-Fard, Ola Svensson, and Rico Zenklusen. Submodular Maximization Subject to Matroid Intersection on the Fly. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 52:1-52:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{feldman_et_al:LIPIcs.ESA.2022.52,
  author =	{Feldman, Moran and Norouzi-Fard, Ashkan and Svensson, Ola and Zenklusen, Rico},
  title =	{{Submodular Maximization Subject to Matroid Intersection on the Fly}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{52:1--52:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.52},
  URN =		{urn:nbn:de:0030-drops-169902},
  doi =		{10.4230/LIPIcs.ESA.2022.52},
  annote =	{Keywords: Submodular Maximization, Matroid Intersection, Streaming Algorithms}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.59
Streaming Submodular Maximization Under Matroid Constraints

Authors: Moran Feldman, Paul Liu, Ashkan Norouzi-Fard, Ola Svensson, and Rico Zenklusen

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Abstract
Recent progress in (semi-)streaming algorithms for monotone submodular function maximization has led to tight results for a simple cardinality constraint. However, current techniques fail to give a similar understanding for natural generalizations, including matroid constraints. This paper aims at closing this gap. For a single matroid of rank k (i.e., any solution has cardinality at most k), our main results are: - A single-pass streaming algorithm that uses Õ(k) memory and achieves an approximation guarantee of 0.3178. - A multi-pass streaming algorithm that uses Õ(k) memory and achieves an approximation guarantee of (1-1/e - ε) by taking a constant (depending on ε) number of passes over the stream. This improves on the previously best approximation guarantees of 1/4 and 1/2 for single-pass and multi-pass streaming algorithms, respectively. In fact, our multi-pass streaming algorithm is tight in that any algorithm with a better guarantee than 1/2 must make several passes through the stream and any algorithm that beats our guarantee of 1-1/e must make linearly many passes (as well as an exponential number of value oracle queries). Moreover, we show how the approach we use for multi-pass streaming can be further strengthened if the elements of the stream arrive in uniformly random order, implying an improved result for p-matchoid constraints.
Cite as

Moran Feldman, Paul Liu, Ashkan Norouzi-Fard, Ola Svensson, and Rico Zenklusen. Streaming Submodular Maximization Under Matroid Constraints. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 59:1-59:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{feldman_et_al:LIPIcs.ICALP.2022.59,
  author =	{Feldman, Moran and Liu, Paul and Norouzi-Fard, Ashkan and Svensson, Ola and Zenklusen, Rico},
  title =	{{Streaming Submodular Maximization Under Matroid Constraints}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{59:1--59:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.59},
  URN =		{urn:nbn:de:0030-drops-164007},
  doi =		{10.4230/LIPIcs.ICALP.2022.59},
  annote =	{Keywords: Submodular maximization, streaming, matroid, random order}
}
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