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Documents authored by Kumar, Anant

Document
DOI: 10.4230/LIPIcs.FSTTCS.2024.20
The Isomorphism Problem of Power Graphs and a Question of Cameron

Authors: Bireswar Das, Jinia Ghosh, and Anant Kumar

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

Abstract
We study the isomorphism problem of graphs that are defined in terms of groups, namely power graphs, directed power graphs, and enhanced power graphs. We design polynomial-time algorithms for the isomorphism problems for the power graphs, the directed power graphs and the enhanced power graphs arising from finite nilpotent groups. In contrast, no polynomial-time algorithm is known for the group isomorphism problem, even for nilpotent groups of class 2. Our algorithms do not require the underlying groups of the input graphs to be given. A crucial step in our algorithms is to reconstruct the directed power graph from the given power graph or the enhanced power graph. The problem of efficiently computing the directed power graph from a power graph or an enhanced power graph is due to Cameron [IJGT'22]. Bubboloni and Pinzauti [Arxiv'22] gave a polynomial-time algorithm to reconstruct the directed power graph from a power graph. We give an efficient algorithm to compute the directed power graph from an enhanced power graph. The tools and techniques that we design are general enough to give a different algorithm to compute the directed power graph from a power graph as well.
Cite as

Bireswar Das, Jinia Ghosh, and Anant Kumar. The Isomorphism Problem of Power Graphs and a Question of Cameron. 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. 20:1-20:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{das_et_al:LIPIcs.FSTTCS.2024.20,
  author =	{Das, Bireswar and Ghosh, Jinia and Kumar, Anant},
  title =	{{The Isomorphism Problem of Power Graphs and a Question of Cameron}},
  booktitle =	{44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024)},
  pages =	{20:1--20:23},
  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.2024.20},
  URN =		{urn:nbn:de:0030-drops-222095},
  doi =		{10.4230/LIPIcs.FSTTCS.2024.20},
  annote =	{Keywords: Graph Isomorphism, Graphs defined on Groups, Power Graphs, Enhanced Power Graphs, Directed Power Graphs, Nilpotent Groups}
}
Document
DOI: 10.4230/LIPIcs.STACS.2023.40
Finding and Counting Patterns in Sparse Graphs

Authors: Balagopal Komarath, Anant Kumar, Suchismita Mishra, and Aditi Sethia

Published in: LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

Abstract
We consider algorithms for finding and counting small, fixed graphs in sparse host graphs. In the non-sparse setting, the parameters treedepth and treewidth play a crucial role in fast, constant-space and polynomial-space algorithms respectively. We discover two new parameters that we call matched treedepth and matched treewidth. We show that finding and counting patterns with low matched treedepth and low matched treewidth can be done asymptotically faster than the existing algorithms when the host graphs are sparse for many patterns. As an application to finding and counting fixed-size patterns, we discover Õ(m³)-time, constant-space algorithms for cycles of length at most 11 and Õ(m²)-time, polynomial-space algorithms for paths of length at most 10.
Cite as

Balagopal Komarath, Anant Kumar, Suchismita Mishra, and Aditi Sethia. Finding and Counting Patterns in Sparse Graphs. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 40:1-40:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{komarath_et_al:LIPIcs.STACS.2023.40,
  author =	{Komarath, Balagopal and Kumar, Anant and Mishra, Suchismita and Sethia, Aditi},
  title =	{{Finding and Counting Patterns in Sparse Graphs}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{40:1--40:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-266-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{254},
  editor =	{Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.40},
  URN =		{urn:nbn:de:0030-drops-176921},
  doi =		{10.4230/LIPIcs.STACS.2023.40},
  annote =	{Keywords: Subgraph Detection and Counting, Homomorphism Polynomials, Treewidth and Treedepth, Matchings}
}
Document
DOI: 10.4230/LIPIcs.STACS.2022.25
Linear Space Data Structures for Finite Groups with Constant Query-Time

Authors: Bireswar Das, Anant Kumar, Shivdutt Sharma, and Dhara Thakkar

Published in: LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Abstract
A finite group of order n can be represented by its Cayley table. In the word-RAM model the Cayley table of a group of order n can be stored using O(n²) words and can be used to answer a multiplication query in constant time. It is interesting to ask if we can design a data structure to store a group of order n that uses o(n²) space but can still answer a multiplication query in constant time. We design a constant query-time data structure that can store any finite group using O(n) words where n is the order of the group. Farzan and Munro (ISSAC 2006) gave an information theoretic lower bound of Ω(n) on the number of words to store a group of order n. Since our data structure achieves this lower bound and answers queries in constant time, it is optimal in both space usage and query-time. A crucial step in the process is essentially to design linear space and constant query-time data structures for nonabelian simple groups. The data structures for nonableian simple groups are designed using a lemma that we prove using the Classification Theorem for Finite Simple Groups (CFSG).
Cite as

Bireswar Das, Anant Kumar, Shivdutt Sharma, and Dhara Thakkar. Linear Space Data Structures for Finite Groups with Constant Query-Time. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{das_et_al:LIPIcs.STACS.2022.25,
  author =	{Das, Bireswar and Kumar, Anant and Sharma, Shivdutt and Thakkar, Dhara},
  title =	{{Linear Space Data Structures for Finite Groups with Constant Query-Time}},
  booktitle =	{39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
  pages =	{25:1--25:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-222-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{219},
  editor =	{Berenbrink, Petra and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.25},
  URN =		{urn:nbn:de:0030-drops-158350},
  doi =		{10.4230/LIPIcs.STACS.2022.25},
  annote =	{Keywords: Compact Data Structures, Space Efficient Representations, Finite Groups, Simple Groups, Classification Theorem for Finite Simple Groups}
}
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