4 Search Results for "Dubey, Chandan"

Document
DOI: 10.4230/LIPIcs.GD.2025.16
Structural Parameterizations of k-Planarity

Authors: Tatsuya Gima, Yasuaki Kobayashi, and Yuto Okada

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract
The concept of k-planarity is extensively studied in the context of Beyond Planarity. A graph is k-planar if it admits a drawing in the plane in which each edge is crossed at most k times. The local crossing number of a graph is the minimum integer k such that it is k-planar. The problem of determining whether an input graph is 1-planar is known to be NP-complete even for near-planar graphs [Cabello and Mohar, SIAM J. Comput. 2013], that is, the graphs obtained from planar graphs by adding a single edge. Moreover, the local crossing number is hard to approximate within a factor 2 - ε for any ε > 0 [Urschel and Wellens, IPL 2021]. To address this computational intractability, Bannister, Cabello, and Eppstein [JGAA 2018] investigated the parameterized complexity of the case of k = 1, particularly focusing on structural parameterizations on input graphs, such as treedepth, vertex cover number, and feedback edge number. In this paper, we extend their approach by considering the general case k ≥ 1 and give (tight) parameterized upper and lower bound results. In particular, we strengthen the aforementioned lower bound results to subclasses of constant-treewidth graphs: we show that testing 1-planarity is NP-complete even for near-planar graphs with feedback vertex set number at most 3 and pathwidth at most 4, and the local crossing number is hard to approximate within any constant factor for graphs with feedback vertex set number at most 2.
Cite as

Tatsuya Gima, Yasuaki Kobayashi, and Yuto Okada. Structural Parameterizations of k-Planarity. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 16:1-16:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{gima_et_al:LIPIcs.GD.2025.16,
  author =	{Gima, Tatsuya and Kobayashi, Yasuaki and Okada, Yuto},
  title =	{{Structural Parameterizations of k-Planarity}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{16:1--16:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.16},
  URN =		{urn:nbn:de:0030-drops-250021},
  doi =		{10.4230/LIPIcs.GD.2025.16},
  annote =	{Keywords: 1-planar graphs, local crossing number, beyond planarity, parameterized complexity, kernelization}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.69
Bandwidth vs BFS Width in Matrix Reordering, Graph Reconstruction, and Graph Drawing

Authors: David Eppstein, Michael T. Goodrich, and Songyu (Alfred) Liu

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
We provide the first approximation quality guarantees for the Cuthull-McKee heuristic for reordering symmetric matrices to have low bandwidth, and we provide an algorithm for reconstructing bounded-bandwidth graphs from distance oracles with near-linear query complexity. To prove these results we introduce a new width parameter, BFS width, and we prove polylogarithmic upper and lower bounds on the BFS width of graphs of bounded bandwidth. Unlike other width parameters, such as bandwidth, pathwidth, and treewidth, BFS width can easily be computed in polynomial time. Bounded BFS width implies bounded bandwidth, pathwidth, and treewidth, which in turn imply fixed-parameter tractable algorithms for many problems that are NP-hard for general graphs. In addition to their applications to matrix ordering, we also provide applications of BFS width to graph reconstruction, to reconstruct graphs from distance queries, and graph drawing, to construct arc diagrams of small height.
Cite as

David Eppstein, Michael T. Goodrich, and Songyu (Alfred) Liu. Bandwidth vs BFS Width in Matrix Reordering, Graph Reconstruction, and Graph Drawing. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 69:1-69:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{eppstein_et_al:LIPIcs.ESA.2025.69,
  author =	{Eppstein, David and Goodrich, Michael T. and Liu, Songyu (Alfred)},
  title =	{{Bandwidth vs BFS Width in Matrix Reordering, Graph Reconstruction, and Graph Drawing}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{69:1--69:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian 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.2025.69},
  URN =		{urn:nbn:de:0030-drops-245373},
  doi =		{10.4230/LIPIcs.ESA.2025.69},
  annote =	{Keywords: Graph algorithms, graph theory, graph width, bandwidth, treewidth}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.65
Recognizing 2-Layer and Outer k-Planar Graphs

Authors: Yasuaki Kobayashi, Yuto Okada, and Alexander Wolff

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
The crossing number of a graph is the least number of crossings over all drawings of the graph in the plane. Computing the crossing number of a given graph is NP-hard, but fixed-parameter tractable (FPT) with respect to the natural parameter. Two well-known variants of the problem are 2-layer crossing minimization and circular crossing minimization, where every vertex must lie on one of two layers, namely two parallel lines, or a circle, respectively. In both cases, edges are drawn as straight-line segments. Both variants are NP-hard, but admit FPT-algorithms with respect to the natural parameter. In recent years, in the context of beyond-planar graphs, a local version of the crossing number has also received considerable attention. A graph is k-planar if it admits a drawing with at most k crossings per edge. In contrast to the crossing number, recognizing k-planar graphs is NP-hard even if k = 1 and hence not likely to be FPT with respect to the natural parameter k. In this paper, we consider the two above variants in the local setting. The k-planar graphs that admit a straight-line drawing with vertices on two layers or on a circle are called 2-layer k-planar and outer k-planar graphs, respectively. We study the parameterized complexity of the two recognition problems with respect to the natural parameter k. For k = 0, the two classes of graphs are exactly the caterpillars and outerplanar graphs, respectively, which can be recognized in linear time. Two groups of researchers independently showed that outer 1-planar graphs can also be recognized in linear time [Hong et al., Algorithmica 2015; Auer et al., Algorithmica 2016]. One group asked explicitly whether outer 2-planar graphs can be recognized in polynomial time. Our main contribution consists of XP-algorithms for recognizing 2-layer k-planar graphs and outer k-planar graphs, which implies that both recognition problems can be solved in polynomial time for every fixed k. We complement these results by showing that recognizing 2-layer k-planar graphs is XNLP-complete and that recognizing outer k-planar graphs is XNLP-hard. This implies that both problems are W[t]-hard for every t and that it is unlikely that they admit FPT-algorithms. On the other hand, we present an FPT-algorithm for recognizing 2-layer k-planar graphs where the order of the vertices on one layer is specified.
Cite as

Yasuaki Kobayashi, Yuto Okada, and Alexander Wolff. Recognizing 2-Layer and Outer k-Planar Graphs. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 65:1-65:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kobayashi_et_al:LIPIcs.SoCG.2025.65,
  author =	{Kobayashi, Yasuaki and Okada, Yuto and Wolff, Alexander},
  title =	{{Recognizing 2-Layer and Outer k-Planar Graphs}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{65:1--65:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.65},
  URN =		{urn:nbn:de:0030-drops-232170},
  doi =		{10.4230/LIPIcs.SoCG.2025.65},
  annote =	{Keywords: 2-layer k-planar graphs, outer k-planar graphs, recognition algorithms, local crossing number, bandwidth, FPT, XNLP, XP, W\lbrackt\rbrack}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2014.643
Sampling a Uniform Solution of a Quadratic Equation Modulo a Prime Power

Authors: Chandan Dubey and Thomas Holenstein

Published in: LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

Abstract
Let p be a prime and k, t be positive integers. Given a quadratic equation Q(x1,x2,...,xn)=t mod p^k in n-variables; we present a polynomial time Las-Vegas algorithm that samples a uniformly random solution of the quadratic equation.
Cite as

Chandan Dubey and Thomas Holenstein. Sampling a Uniform Solution of a Quadratic Equation Modulo a Prime Power. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 643-653, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{dubey_et_al:LIPIcs.APPROX-RANDOM.2014.643,
  author =	{Dubey, Chandan and Holenstein, Thomas},
  title =	{{Sampling a Uniform Solution of a Quadratic Equation Modulo a Prime Power}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)},
  pages =	{643--653},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-74-3},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{28},
  editor =	{Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.643},
  URN =		{urn:nbn:de:0030-drops-47289},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2014.643},
  annote =	{Keywords: Quadratic Forms, Lattices, Modular, p-adic}
}
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