6 Search Results for "Kapralov, Michael"

Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2023.57
On Constructing Spanners from Random Gaussian Projections

Authors: Sepehr Assadi, Michael Kapralov, and Huacheng Yu

Published in: LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

Abstract
Graph sketching is a powerful paradigm for analyzing graph structure via linear measurements introduced by Ahn, Guha, and McGregor (SODA'12) that has since found numerous applications in streaming, distributed computing, and massively parallel algorithms, among others. Graph sketching has proven to be quite successful for various problems such as connectivity, minimum spanning trees, edge or vertex connectivity, and cut or spectral sparsifiers. Yet, the problem of approximating shortest path metric of a graph, and specifically computing a spanner, is notably missing from the list of successes. This has turned the status of this fundamental problem into one of the most longstanding open questions in this area. We present a partial explanation of this lack of success by proving a strong lower bound for a large family of graph sketching algorithms that encompasses prior work on spanners and many (but importantly not also all) related cut-based problems mentioned above. Our lower bound matches the algorithmic bounds of the recent result of Filtser, Kapralov, and Nouri (SODA'21), up to lower order terms, for constructing spanners via the same graph sketching family. This establishes near-optimality of these bounds, at least restricted to this family of graph sketching techniques, and makes progress on a conjecture posed in this latter work.
Cite as

Sepehr Assadi, Michael Kapralov, and Huacheng Yu. On Constructing Spanners from Random Gaussian Projections. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 57:1-57:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{assadi_et_al:LIPIcs.APPROX/RANDOM.2023.57,
  author =	{Assadi, Sepehr and Kapralov, Michael and Yu, Huacheng},
  title =	{{On Constructing Spanners from Random Gaussian Projections}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{57:1--57:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-296-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{275},
  editor =	{Megow, Nicole and Smith, Adam},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.57},
  URN =		{urn:nbn:de:0030-drops-188821},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.57},
  annote =	{Keywords: sketching algorithm, lower bound, graph spanner}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2023.50
Expander Decomposition in Dynamic Streams

Authors: Arnold Filtser, Michael Kapralov, and Mikhail Makarov

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Abstract
In this paper we initiate the study of expander decompositions of a graph G = (V, E) in the streaming model of computation. The goal is to find a partitioning 𝒞 of vertices V such that the subgraphs of G induced by the clusters C ∈ 𝒞 are good expanders, while the number of intercluster edges is small. Expander decompositions are classically constructed by a recursively applying balanced sparse cuts to the input graph. In this paper we give the first implementation of such a recursive sparsest cut process using small space in the dynamic streaming model. Our main algorithmic tool is a new type of cut sparsifier that we refer to as a power cut sparsifier - it preserves cuts in any given vertex induced subgraph (or, any cluster in a fixed partition of V) to within a (δ, ε)-multiplicative/additive error with high probability. The power cut sparsifier uses Õ(n/εδ) space and edges, which we show is asymptotically tight up to polylogarithmic factors in n for constant δ.
Cite as

Arnold Filtser, Michael Kapralov, and Mikhail Makarov. Expander Decomposition in Dynamic Streams. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 50:1-50:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{filtser_et_al:LIPIcs.ITCS.2023.50,
  author =	{Filtser, Arnold and Kapralov, Michael and Makarov, Mikhail},
  title =	{{Expander Decomposition in Dynamic Streams}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{50:1--50:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.50},
  URN =		{urn:nbn:de:0030-drops-175534},
  doi =		{10.4230/LIPIcs.ITCS.2023.50},
  annote =	{Keywords: Streaming, expander decomposition, graph sparsifiers}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.91
Noisy Boolean Hidden Matching with Applications

Authors: Michael Kapralov, Amulya Musipatla, Jakab Tardos, David P. Woodruff, and Samson Zhou

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract
The Boolean Hidden Matching (BHM) problem, introduced in a seminal paper of Gavinsky et al. [STOC'07], has played an important role in lower bounds for graph problems in the streaming model (e.g., subgraph counting, maximum matching, MAX-CUT, Schatten p-norm approximation). The BHM problem typically leads to Ω(√n) space lower bounds for constant factor approximations, with the reductions generating graphs that consist of connected components of constant size. The related Boolean Hidden Hypermatching (BHH) problem provides Ω(n^{1-1/t}) lower bounds for 1+O(1/t) approximation, for integers t ≥ 2. The corresponding reductions produce graphs with connected components of diameter about t, and essentially show that long range exploration is hard in the streaming model with an adversarial order of updates. In this paper we introduce a natural variant of the BHM problem, called noisy BHM (and its natural noisy BHH variant), that we use to obtain stronger than Ω(√n) lower bounds for approximating a number of the aforementioned problems in graph streams when the input graphs consist only of components of diameter bounded by a fixed constant. We next introduce and study the graph classification problem, where the task is to test whether the input graph is isomorphic to a given graph. As a first step, we use the noisy BHM problem to show that the problem of classifying whether an underlying graph is isomorphic to a complete binary tree in insertion-only streams requires Ω(n) space, which seems challenging to show using either BHM or BHH.
Cite as

Michael Kapralov, Amulya Musipatla, Jakab Tardos, David P. Woodruff, and Samson Zhou. Noisy Boolean Hidden Matching with Applications. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 91:1-91:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kapralov_et_al:LIPIcs.ITCS.2022.91,
  author =	{Kapralov, Michael and Musipatla, Amulya and Tardos, Jakab and Woodruff, David P. and Zhou, Samson},
  title =	{{Noisy Boolean Hidden Matching with Applications}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{91:1--91:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.91},
  URN =		{urn:nbn:de:0030-drops-156876},
  doi =		{10.4230/LIPIcs.ITCS.2022.91},
  annote =	{Keywords: Boolean Hidden Matching, Lower Bounds, Communication Complexity, Streaming Algorithms}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2019.6
A Simple Sublinear-Time Algorithm for Counting Arbitrary Subgraphs via Edge Sampling

Authors: Sepehr Assadi, Michael Kapralov, and Sanjeev Khanna

Published in: LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

Abstract
In the subgraph counting problem, we are given a (large) input graph G(V, E) and a (small) target graph H (e.g., a triangle); the goal is to estimate the number of occurrences of H in G. Our focus here is on designing sublinear-time algorithms for approximately computing number of occurrences of H in G in the setting where the algorithm is given query access to G. This problem has been studied in several recent papers which primarily focused on specific families of graphs H such as triangles, cliques, and stars. However, not much is known about approximate counting of arbitrary graphs H in the literature. This is in sharp contrast to the closely related subgraph enumeration problem that has received significant attention in the database community as the database join problem. The AGM bound shows that the maximum number of occurrences of any arbitrary subgraph H in a graph G with m edges is O(m^{rho(H)}), where rho(H) is the fractional edge-cover of H, and enumeration algorithms with matching runtime are known for any H. We bridge this gap between subgraph counting and subgraph enumeration by designing a simple sublinear-time algorithm that can estimate the number of occurrences of any arbitrary graph H in G, denoted by #H, to within a (1 +/- epsilon)-approximation with high probability in O(m^{rho(H)}/#H) * poly(log(n),1/epsilon) time. Our algorithm is allowed the standard set of queries for general graphs, namely degree queries, pair queries and neighbor queries, plus an additional edge-sample query that returns an edge chosen uniformly at random. The performance of our algorithm matches those of Eden et al. [FOCS 2015, STOC 2018] for counting triangles and cliques and extend them to all choices of subgraph H under the additional assumption of edge-sample queries.
Cite as

Sepehr Assadi, Michael Kapralov, and Sanjeev Khanna. A Simple Sublinear-Time Algorithm for Counting Arbitrary Subgraphs via Edge Sampling. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 6:1-6:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{assadi_et_al:LIPIcs.ITCS.2019.6,
  author =	{Assadi, Sepehr and Kapralov, Michael and Khanna, Sanjeev},
  title =	{{A Simple Sublinear-Time Algorithm for Counting Arbitrary Subgraphs via Edge Sampling}},
  booktitle =	{10th Innovations in Theoretical Computer Science Conference (ITCS 2019)},
  pages =	{6:1--6:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-095-8},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{124},
  editor =	{Blum, Avrim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.6},
  URN =		{urn:nbn:de:0030-drops-100996},
  doi =		{10.4230/LIPIcs.ITCS.2019.6},
  annote =	{Keywords: Sublinear-time algorithms, Subgraph counting, AGM bound}
}
Document
DOI: 10.4230/OASIcs.SOSA.2019.11
Towards a Unified Theory of Sparsification for Matching Problems

Authors: Sepehr Assadi and Aaron Bernstein

Published in: OASIcs, Volume 69, 2nd Symposium on Simplicity in Algorithms (SOSA 2019)

Abstract
In this paper, we present a construction of a "matching sparsifier", that is, a sparse subgraph of the given graph that preserves large matchings approximately and is robust to modifications of the graph. We use this matching sparsifier to obtain several new algorithmic results for the maximum matching problem: - An almost (3/2)-approximation one-way communication protocol for the maximum matching problem, significantly simplifying the (3/2)-approximation protocol of Goel, Kapralov, and Khanna (SODA 2012) and extending it from bipartite graphs to general graphs. - An almost (3/2)-approximation algorithm for the stochastic matching problem, improving upon and significantly simplifying the previous 1.999-approximation algorithm of Assadi, Khanna, and Li (EC 2017). - An almost (3/2)-approximation algorithm for the fault-tolerant matching problem, which, to our knowledge, is the first non-trivial algorithm for this problem. Our matching sparsifier is obtained by proving new properties of the edge-degree constrained subgraph (EDCS) of Bernstein and Stein (ICALP 2015; SODA 2016) - designed in the context of maintaining matchings in dynamic graphs - that identifies EDCS as an excellent choice for a matching sparsifier. This leads to surprisingly simple and non-technical proofs of the above results in a unified way. Along the way, we also provide a much simpler proof of the fact that an EDCS is guaranteed to contain a large matching, which may be of independent interest.
Cite as

Sepehr Assadi and Aaron Bernstein. Towards a Unified Theory of Sparsification for Matching Problems. In 2nd Symposium on Simplicity in Algorithms (SOSA 2019). Open Access Series in Informatics (OASIcs), Volume 69, pp. 11:1-11:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{assadi_et_al:OASIcs.SOSA.2019.11,
  author =	{Assadi, Sepehr and Bernstein, Aaron},
  title =	{{Towards a Unified Theory of Sparsification for Matching Problems}},
  booktitle =	{2nd Symposium on Simplicity in Algorithms (SOSA 2019)},
  pages =	{11:1--11:20},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-099-6},
  ISSN =	{2190-6807},
  year =	{2019},
  volume =	{69},
  editor =	{Fineman, Jeremy T. and Mitzenmacher, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2019.11},
  URN =		{urn:nbn:de:0030-drops-100370},
  doi =		{10.4230/OASIcs.SOSA.2019.11},
  annote =	{Keywords: Maximum matching, matching sparsifiers, one-way communication complexity, stochastic matching, fault-tolerant matching}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2016.7
Online Row Sampling

Authors: Michael B. Cohen, Cameron Musco, and Jakub Pachocki

Published in: LIPIcs, Volume 60, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)

Abstract
Finding a small spectral approximation for a tall n x d matrix A is a fundamental numerical primitive. For a number of reasons, one often seeks an approximation whose rows are sampled from those of A. Row sampling improves interpretability, saves space when A is sparse, and preserves row structure, which is especially important, for example, when A represents a graph. However, correctly sampling rows from A can be costly when the matrix is large and cannot be stored and processed in memory. Hence, a number of recent publications focus on row sampling in the streaming setting, using little more space than what is required to store the outputted approximation [Kelner Levin 2013] [Kapralov et al. 2014]. Inspired by a growing body of work on online algorithms for machine learning and data analysis, we extend this work to a more restrictive online setting: we read rows of A one by one and immediately decide whether each row should be kept in the spectral approximation or discarded, without ever retracting these decisions. We present an extremely simple algorithm that approximates A up to multiplicative error epsilon and additive error delta using O(d log d log (epsilon ||A||_2^2/delta) / epsilon^2) online samples, with memory overhead proportional to the cost of storing the spectral approximation. We also present an algorithm that uses O(d^2) memory but only requires O(d log (epsilon ||A||_2^2/delta) / epsilon^2) samples, which we show is optimal. Our methods are clean and intuitive, allow for lower memory usage than prior work, and expose new theoretical properties of leverage score based matrix approximation.
Cite as

Michael B. Cohen, Cameron Musco, and Jakub Pachocki. Online Row Sampling. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 7:1-7:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{cohen_et_al:LIPIcs.APPROX-RANDOM.2016.7,
  author =	{Cohen, Michael B. and Musco, Cameron and Pachocki, Jakub},
  title =	{{Online Row Sampling}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)},
  pages =	{7:1--7:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-018-7},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{60},
  editor =	{Jansen, Klaus and Mathieu, Claire and Rolim, Jos\'{e} D. P. and Umans, Chris},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2016.7},
  URN =		{urn:nbn:de:0030-drops-66304},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2016.7},
  annote =	{Keywords: spectral sparsification, leverage score sampling, online sparsification}
}
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