Document

Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

We study the problem of maximizing a non-monotone submodular function subject to a cardinality constraint in the streaming model. Our main contributions are two single-pass (semi-)streaming algorithms that use Õ(k)⋅poly(1/ε) memory, where k is the size constraint. At the end of the stream, both our algorithms post-process their data structures using any offline algorithm for submodular maximization, and obtain a solution whose approximation guarantee is α/(1+α)-ε, where α is the approximation of the offline algorithm. If we use an exact (exponential time) post-processing algorithm, this leads to 1/2-ε approximation (which is nearly optimal). If we post-process with the algorithm of [Niv Buchbinder and Moran Feldman, 2019], that achieves the state-of-the-art offline approximation guarantee of α = 0.385, we obtain 0.2779-approximation in polynomial time, improving over the previously best polynomial-time approximation of 0.1715 due to [Feldman et al., 2018]. One of our algorithms is combinatorial and enjoys fast update and overall running times. Our other algorithm is based on the multilinear extension, enjoys an improved space complexity, and can be made deterministic in some settings of interest.

Naor Alaluf, Alina Ene, Moran Feldman, Huy L. Nguyen, and Andrew Suh. Optimal Streaming Algorithms for Submodular Maximization with Cardinality Constraints. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 6:1-6:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{alaluf_et_al:LIPIcs.ICALP.2020.6, author = {Alaluf, Naor and Ene, Alina and Feldman, Moran and Nguyen, Huy L. and Suh, Andrew}, title = {{Optimal Streaming Algorithms for Submodular Maximization with Cardinality Constraints}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {6:1--6:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.6}, URN = {urn:nbn:de:0030-drops-124137}, doi = {10.4230/LIPIcs.ICALP.2020.6}, annote = {Keywords: Submodular maximization, streaming algorithms, cardinality constraint} }

Document

Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

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.

Alina Ene and Huy L. Nguyen. A Nearly-Linear Time Algorithm for Submodular Maximization with a Knapsack Constraint. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 53:1-53:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{ene_et_al:LIPIcs.ICALP.2019.53, author = {Ene, Alina and Nguyen, Huy L.}, 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 = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.53}, URN = {urn:nbn:de:0030-drops-106290}, doi = {10.4230/LIPIcs.ICALP.2019.53}, annote = {Keywords: submodular maximization, knapsack constraint, fast algorithms} }

Document

Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

We consider fast algorithms for monotone submodular maximization subject to a matroid constraint. We assume that the matroid is given as input in an explicit form, and the goal is to obtain the best possible running times for important matroids. We develop a new algorithm for a general matroid constraint with a 1 - 1/e - epsilon approximation that achieves a fast running time provided we have a fast data structure for maintaining an approximately maximum weight base in the matroid through a sequence of decrease weight operations. We construct such data structures for graphic matroids and partition matroids, and we obtain the first algorithms for these classes of matroids that achieve a nearly-optimal, 1 - 1/e - epsilon approximation, using a nearly-linear number of function evaluations and arithmetic operations.

Alina Ene and Huy L. Nguyen. Towards Nearly-Linear Time Algorithms for Submodular Maximization with a Matroid Constraint. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 54:1-54:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{ene_et_al:LIPIcs.ICALP.2019.54, author = {Ene, Alina and Nguyen, Huy L.}, title = {{Towards Nearly-Linear Time Algorithms for Submodular Maximization with a Matroid Constraint}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {54:1--54:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.54}, URN = {urn:nbn:de:0030-drops-106303}, doi = {10.4230/LIPIcs.ICALP.2019.54}, annote = {Keywords: submodular maximization, matroid constraints, fast running times} }