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Documents authored by Sengupta, Rik

Document
DOI: 10.4230/LIPIcs.FSTTCS.2024.29
Matchings in Low-Arboricity Graphs in the Dynamic Graph Stream Model

Authors: Christian Konrad, Andrew McGregor, Rik Sengupta, and Cuong Than

Published in: LIPIcs, Volume 323, 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024)

Abstract
We consider the problem of estimating the size of a maximum matching in low-arboricity graphs in the dynamic graph stream model. In this setting, an algorithm with limited memory makes multiple passes over a stream of edge insertions and deletions, resulting in a low-arboricity graph. Let n be the number of vertices of the input graph, and α be its arboricity. We give the following results. 1) As our main result, we give a three-pass streaming algorithm that produces an (α + 2)(1 + ε)-approximation and uses space O(ε^{-2}⋅α²⋅n^{1/2}⋅log n). This result should be contrasted with the Ω(α^{-5/2}⋅n^{1/2}) space lower bound established by [Assadi et al., SODA'17] for one-pass algorithms, showing that, for graphs of constant arboricity, the one-pass space lower bound can be achieved in three passes (up to poly-logarithmic factors). Furthermore, we obtain a two-pass algorithm that uses space O(ε^{-2}⋅α²⋅n^{3/5}⋅log n). 2) We also give a (1+ε)-approximation multi-pass algorithm, where the space used is parameterized by an upper bound on the size of a largest matching. For example, using O(log log n) passes, the space required is O(ε^{-1}⋅α²⋅k⋅log n), where k denotes an upper bound on the size of a largest matching. Finally, we define a notion of arboricity in the context of matrices. This is a natural measure of the sparsity of a matrix that is more nuanced than simply bounding the total number of nonzero entries, but less restrictive than bounding the number of nonzero entries in each row and column. For such matrices, we exploit our results on estimating matching size to present upper bounds for the problem of rank estimation in the dynamic data stream model.
Cite as

Christian Konrad, Andrew McGregor, Rik Sengupta, and Cuong Than. Matchings in Low-Arboricity Graphs in the Dynamic Graph Stream Model. In 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 323, pp. 29:1-29:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{konrad_et_al:LIPIcs.FSTTCS.2024.29,
  author =	{Konrad, Christian and McGregor, Andrew and Sengupta, Rik and Than, Cuong},
  title =	{{Matchings in Low-Arboricity Graphs in the Dynamic Graph Stream Model}},
  booktitle =	{44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024)},
  pages =	{29:1--29:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-355-3},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{323},
  editor =	{Barman, Siddharth and Lasota, S{\l}awomir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2024.29},
  URN =		{urn:nbn:de:0030-drops-222187},
  doi =		{10.4230/LIPIcs.FSTTCS.2024.29},
  annote =	{Keywords: Data Streams, Graph Matching, Graph Arboricity}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2024.34
On the Number of Quantifiers Needed to Define Boolean Functions

Authors: Marco Carmosino, Ronald Fagin, Neil Immerman, Phokion G. Kolaitis, Jonathan Lenchner, and Rik Sengupta

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract
The number of quantifiers needed to express first-order (FO) properties is captured by two-player combinatorial games called multi-structural games. We analyze these games on binary strings with an ordering relation, using a technique we call parallel play, which significantly reduces the number of quantifiers needed in many cases. Ordered structures such as strings have historically been notoriously difficult to analyze in the context of these and similar games. Nevertheless, in this paper, we provide essentially tight bounds on the number of quantifiers needed to characterize different-sized subsets of strings. The results immediately give bounds on the number of quantifiers necessary to define several different classes of Boolean functions. One of our results is analogous to Lupanov’s upper bounds on circuit size and formula size in propositional logic: we show that every Boolean function on n-bit inputs can be defined by a FO sentence having (1+ε)n/log(n) + O(1) quantifiers, and that this is essentially tight. We reduce this number to (1 + ε)log(n) + O(1) when the Boolean function in question is sparse.
Cite as

Marco Carmosino, Ronald Fagin, Neil Immerman, Phokion G. Kolaitis, Jonathan Lenchner, and Rik Sengupta. On the Number of Quantifiers Needed to Define Boolean Functions. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 34:1-34:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{carmosino_et_al:LIPIcs.MFCS.2024.34,
  author =	{Carmosino, Marco and Fagin, Ronald and Immerman, Neil and Kolaitis, Phokion G. and Lenchner, Jonathan and Sengupta, Rik},
  title =	{{On the Number of Quantifiers Needed to Define Boolean Functions}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{34:1--34:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.34},
  URN =		{urn:nbn:de:0030-drops-205907},
  doi =		{10.4230/LIPIcs.MFCS.2024.34},
  annote =	{Keywords: logic, combinatorial games, Boolean functions, quantifier number}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.96
Graph Reconstruction from Random Subgraphs

Authors: Andrew McGregor and Rik Sengupta

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

Abstract
We consider the problem of reconstructing a graph G in two natural sampling models: 1) each sample corresponds to a random induced subgraph and 2) for a fixed adjacency matrix A_G for G, each sample corresponds to a random principal submatrix (i.e., a submatrix formed by deleting the same set of rows and columns) of A_G. We refer to these models as the "unordered" and "ordered" models respectively. The two models are motivated by work on the reconstruction conjecture in combinatorics and trace reconstruction in theoretical computer science. Despite the superficial similarities between the models, we show that the sample complexity of reconstruction can be exponentially different. Our main results are as follows: - In the unordered model, we show that almost all graphs can be reconstructed with Θ(p^{-2} log n) samples if each node is included in the random subgraph with any constant probability p; this is optimal. We show our upper bound extends to smaller values of p as well. In contrast, for arbitrary graphs, we show that exp(Ω(n)) samples are required for reconstruction even for 2-regular graphs. One of the key technical steps in the first result is showing that, with high probability, any subgraph isomorphism in a random graph has at most O(log n) non-fixed points. - In the ordered model, we show that any graph with constant arboricity or degeneracy (i.e., every induced subgraph has constant average degree) can be reconstructed with exp(Õ(n^{1/3})) samples and that arbitrary graphs can be reconstructed with exp(Õ(n^{1/2})) samples. The results about almost all graphs in the first model carry over to the second. One of the key technical steps in the first result is showing that reconstruction of low degeneracy graphs can be reduced to learning a small number of moments of sets of the form {i-j: j < i,(i,j) ∈ E} and {j-i: i < j,(i,j) ∈ E} where G = ([n],E) is the unknown graph.
Cite as

Andrew McGregor and Rik Sengupta. Graph Reconstruction from Random Subgraphs. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 96:1-96:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{mcgregor_et_al:LIPIcs.ICALP.2022.96,
  author =	{McGregor, Andrew and Sengupta, Rik},
  title =	{{Graph Reconstruction from Random Subgraphs}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{96:1--96:18},
  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.2022.96},
  URN =		{urn:nbn:de:0030-drops-164373},
  doi =		{10.4230/LIPIcs.ICALP.2022.96},
  annote =	{Keywords: graph reconstruction, sample complexity, deletion channel}
}
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