4 Search Results for "Sellier, François"


Document
Parameterized Matroid-Constrained Maximum Coverage

Authors: François Sellier

Published in: LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)


Abstract
In this paper, we introduce the concept of Density-Balanced Subset in a matroid, in which independent sets can be sampled so as to guarantee that (i) each element has the same probability to be sampled, and (ii) those events are negatively correlated. These Density-Balanced Subsets are subsets in the ground set of a matroid in which the traditional notion of uniform random sampling can be extended. We then provide an application of this concept to the Matroid-Constrained Maximum Coverage problem. In this problem, given a matroid ℳ = (V, ℐ) of rank k on a ground set V and a coverage function f on V, the goal is to find an independent set S ∈ ℐ maximizing f(S). This problem is an important special case of the much-studied submodular function maximization problem subject to a matroid constraint; this is also a generalization of the maximum k-cover problem in a graph. In this paper, assuming that the coverage function has a bounded frequency μ (i.e., any element of the underlying universe of the coverage function appears in at most μ sets), we design a procedure, parameterized by some integer ρ, to extract in polynomial time an approximate kernel of size ρ ⋅ k that is guaranteed to contain a 1 - (μ - 1)/ρ approximation of the optimal solution. This procedure can then be used to get a Fixed-Parameter Tractable Approximation Scheme (FPT-AS) providing a 1 - ε approximation in time (μ/ε)^O(k) ⋅ |V|^O(1). This generalizes and improves the results of [Manurangsi, 2019] and [Huang and Sellier, 2022], providing the first FPT-AS working on an arbitrary matroid. Moreover, as the kernel has a very simple characterization, it can be constructed in the streaming setting.

Cite as

François Sellier. Parameterized Matroid-Constrained Maximum Coverage. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 94:1-94:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{sellier:LIPIcs.ESA.2023.94,
  author =	{Sellier, Fran\c{c}ois},
  title =	{{Parameterized Matroid-Constrained Maximum Coverage}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{94:1--94:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. 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.2023.94},
  URN =		{urn:nbn:de:0030-drops-187475},
  doi =		{10.4230/LIPIcs.ESA.2023.94},
  annote =	{Keywords: Matroids, approximate kernel, maximum coverage}
}
Document
Maximum Weight b-Matchings in Random-Order Streams

Authors: Chien-Chung Huang and François Sellier

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


Abstract
We consider the maximum weight b-matching problem in the random-order semi-streaming model. Assuming all weights are small integers drawn from [1,W], we present a 2 - 1/(2W) + ε approximation algorithm, using a memory of O(max(|M_G|, n) ⋅ poly(log(m),W,1/ε)), where |M_G| denotes the cardinality of the optimal matching. Our result generalizes that of Bernstein [Aaron Bernstein, 2020], which achieves a 3/2 + ε approximation for the maximum cardinality simple matching. When W is small, our result also improves upon that of Gamlath et al. [Gamlath et al., 2019], which obtains a 2 - δ approximation (for some small constant δ ∼ 10^{-17}) for the maximum weight simple matching. In particular, for the weighted b-matching problem, ours is the first result beating the approximation ratio of 2. Our technique hinges on a generalized weighted version of edge-degree constrained subgraphs, originally developed by Bernstein and Stein [Aaron Bernstein and Cliff Stein, 2015]. Such a subgraph has bounded vertex degree (hence uses only a small number of edges), and can be easily computed. The fact that it contains a 2 - 1/(2W) + ε approximation of the maximum weight matching is proved using the classical Kőnig-Egerváry’s duality theorem.

Cite as

Chien-Chung Huang and François Sellier. Maximum Weight b-Matchings in Random-Order Streams. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 68:1-68:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{huang_et_al:LIPIcs.ESA.2022.68,
  author =	{Huang, Chien-Chung and Sellier, Fran\c{c}ois},
  title =	{{Maximum Weight b-Matchings in Random-Order Streams}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{68:1--68:14},
  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.2022.68},
  URN =		{urn:nbn:de:0030-drops-170062},
  doi =		{10.4230/LIPIcs.ESA.2022.68},
  annote =	{Keywords: Maximum weight matching, b-matching, streaming, random order}
}
Document
Matroid-Constrained Maximum Vertex Cover: Approximate Kernels and Streaming Algorithms

Authors: Chien-Chung Huang and François Sellier

Published in: LIPIcs, Volume 227, 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)


Abstract
Given a graph with weights on the edges and a matroid imposed on the vertices, our problem is to choose a subset of vertices that is independent in the matroid, with the objective of maximizing the total weight of covered edges. This problem is a generalization of the much studied max k-vertex cover problem, where the matroid is the simple uniform matroid, and it is also a special case of maximizing a monotone submodular function under a matroid constraint. In this work, we give a Fixed Parameter Tractable Approximation Scheme (FPT-AS) when the given matroid is a partition matroid, a laminar matroid, or a transversal matroid. Precisely, if k is the rank of the matroid, we obtain (1 - ε) approximation using (1/(ε))^{O(k)}n^{O(1)} time for partition and laminar matroids and using (1/(ε)+k)^{O(k)}n^{O(1)} time for transversal matroids. This extends a result of Manurangsi for uniform matroids [Pasin Manurangsi, 2018]. We also show that these ideas can be applied in the context of (single-pass) streaming algorithms. Our FPT-AS introduces a new technique based on matroid union, which may be of independent interest in extremal combinatorics.

Cite as

Chien-Chung Huang and François Sellier. Matroid-Constrained Maximum Vertex Cover: Approximate Kernels and Streaming Algorithms. In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 27:1-27:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{huang_et_al:LIPIcs.SWAT.2022.27,
  author =	{Huang, Chien-Chung and Sellier, Fran\c{c}ois},
  title =	{{Matroid-Constrained Maximum Vertex Cover: Approximate Kernels and Streaming Algorithms}},
  booktitle =	{18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)},
  pages =	{27:1--27:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-236-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{227},
  editor =	{Czumaj, Artur and Xin, Qin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2022.27},
  URN =		{urn:nbn:de:0030-drops-161874},
  doi =		{10.4230/LIPIcs.SWAT.2022.27},
  annote =	{Keywords: Maximum vertex cover, matroid, approximate kernel, streaming}
}
Document
APPROX
Semi-Streaming Algorithms for Submodular Function Maximization Under b-Matching Constraint

Authors: Chien-Chung Huang and François Sellier

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)


Abstract
We consider the problem of maximizing a submodular function under the b-matching constraint, in the semi-streaming model. Our main results can be summarized as follows. - When the function is linear, i.e. for the maximum weight b-matching problem, we obtain a 2+ε approximation. This improves the previous best bound of 3+ε [Roie Levin and David Wajc, 2021]. - When the function is a non-negative monotone submodular function, we obtain a 3 + 2 √2 ≈ 5.828 approximation. This matches the currently best ratio [Roie Levin and David Wajc, 2021]. - When the function is a non-negative non-monotone submodular function, we obtain a 4 + 2 √3 ≈ 7.464 approximation. This ratio is also achieved in [Roie Levin and David Wajc, 2021], but only under the simple matching constraint, while we can deal with the more general b-matching constraint. We also consider a generalized problem, where a k-uniform hypergraph is given with an extra matroid constraint imposed on the edges, with the same goal of finding a b-matching that maximizes a submodular function. We extend our technique to this case to obtain an algorithm with an approximation of 8/3k+O(1). Our algorithms build on the ideas of the recent works of Levin and Wajc [Roie Levin and David Wajc, 2021] and of Garg, Jordan, and Svensson [Paritosh Garg et al., 2021]. Our main technical innovation is to introduce a data structure and associate it with each vertex and the matroid, to record the extra information of the stored edges. After the streaming phase, these data structures guide the greedy algorithm to make better choices.

Cite as

Chien-Chung Huang and François Sellier. Semi-Streaming Algorithms for Submodular Function Maximization Under b-Matching Constraint. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 14:1-14:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{huang_et_al:LIPIcs.APPROX/RANDOM.2021.14,
  author =	{Huang, Chien-Chung and Sellier, Fran\c{c}ois},
  title =	{{Semi-Streaming Algorithms for Submodular Function Maximization Under b-Matching Constraint}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{14:1--14:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.14},
  URN =		{urn:nbn:de:0030-drops-147072},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.14},
  annote =	{Keywords: Maximum Weight Matching, Submodular Function Maximization, Streaming, Matroid}
}
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