3 Search Results for "Nielsen, Jesper Sindahl"

Document
DOI: 10.4230/LIPIcs.STACS.2026.13
Line Cover and Related Problems

Authors: Matthias Bentert, Fedor V. Fomin, Petr A. Golovach, Souvik Saha, Sanjay Seetharaman, and Anannya Upasana

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract
We study several extensions of the classic Line Cover problem of covering a set of n points in the plane with k lines. Line Cover is known to be NP-hard and our focus is on two natural generalizations: (1) Line Clustering, where the objective is to find k lines in the plane that minimize the sum of squares of distances of a given set of input points to the closest line, and (2) Hyperplane Cover, where the goal is to cover n points in ℝ^d by k hyperplanes. We also consider the more general Projective Clustering problem, which unifies both of these and has numerous applications in machine learning, data mining, and computational geometry. In this problem one seeks k affine subspaces of dimension r minimizing the sum of squares of distances of a given set of n points in ℝ^d to the closest point within one of the k affine subspaces. Our main contributions reveal interesting differences in the parameterized complexity of these problems. While Line Cover is fixed-parameter tractable parameterized by the number k of lines in the solution, we show that Line Clustering is W[1]-hard when parameterized by k and rule out algorithms of running time n^{o(k)} under the Exponential Time Hypothesis. Hyperplane Cover is known to be NP-hard even when d = 2 and by the work of Langerman and Morin [Discrete & Computational Geometry, 2005], it is FPT parameterized by k and d. We complement this result by establishing that Hyperplane Cover is W[2]-hard when parameterized by only k. We complement our hardness results by presenting an algorithm for Projective Clustering. We show that this problem is solvable in n^{𝒪(dk(r+1))} time. Not only does this yield an upper bound for Line Clustering that asymptotically matches our lower bound, but it also significantly extends the seminal work on k-Means Clustering (the special case r = 0) by Inaba, Katoh, and Imai [SoCG 1994].
Cite as

Matthias Bentert, Fedor V. Fomin, Petr A. Golovach, Souvik Saha, Sanjay Seetharaman, and Anannya Upasana. Line Cover and Related Problems. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 13:1-13:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bentert_et_al:LIPIcs.STACS.2026.13,
  author =	{Bentert, Matthias and Fomin, Fedor V. and Golovach, Petr A. and Saha, Souvik and Seetharaman, Sanjay and Upasana, Anannya},
  title =	{{Line Cover and Related Problems}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{13:1--13:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.13},
  URN =		{urn:nbn:de:0030-drops-255023},
  doi =		{10.4230/LIPIcs.STACS.2026.13},
  annote =	{Keywords: Point Line Cover, Projective Clustering, W-hardness, XP algorithm}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2016.93
Data Structure Lower Bounds for Document Indexing Problems

Authors: Peyman Afshani and Jesper Sindahl Nielsen

Published in: LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

Abstract
We study data structure problems related to document indexing and pattern matching queries and our main contribution is to show that the pointer machine model of computation can be extremely useful in proving high and unconditional lower bounds that cannot be obtained in any other known model of computation with the current techniques. Often our lower bounds match the known space-query time trade-off curve and in fact for all the problems considered, there is a very good and reasonable match between our lower bounds and the known upper bounds, at least for some choice of input parameters. The problems that we consider are set intersection queries (both the reporting variant and the semi-group counting variant), indexing a set of documents for two-pattern queries, or forbidden-pattern queries, or queries with wild-cards, and indexing an input set of gapped-patterns (or two-patterns) to find those matching a document given at the query time.
Cite as

Peyman Afshani and Jesper Sindahl Nielsen. Data Structure Lower Bounds for Document Indexing Problems. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 93:1-93:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{afshani_et_al:LIPIcs.ICALP.2016.93,
  author =	{Afshani, Peyman and Nielsen, Jesper Sindahl},
  title =	{{Data Structure Lower Bounds for Document Indexing Problems}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{93:1--93:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.93},
  URN =		{urn:nbn:de:0030-drops-61923},
  doi =		{10.4230/LIPIcs.ICALP.2016.93},
  annote =	{Keywords: Data Structure Lower Bounds, Pointer Machine, Set Intersection, Pattern Matching}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2016.60
Applications of Incidence Bounds in Point Covering Problems

Authors: Peyman Afshani, Edvin Berglin, Ingo van Duijn, and Jesper Sindahl Nielsen

Published in: LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

Abstract
In the Line Cover problem a set of n points is given and the task is to cover the points using either the minimum number of lines or at most k lines. In Curve Cover, a generalization of Line Cover, the task is to cover the points using curves with d degrees of freedom. Another generalization is the Hyperplane Cover problem where points in d-dimensional space are to be covered by hyperplanes. All these problems have kernels of polynomial size, where the parameter is the minimum number of lines, curves, or hyperplanes needed. First we give a non-parameterized algorithm for both problems in O*(2^n) (where the O*(.) notation hides polynomial factors of n) time and polynomial space, beating a previous exponential-space result. Combining this with incidence bounds similar to the famous Szemeredi-Trotter bound, we present a Curve Cover algorithm with running time O*((Ck/log k)^((d-1)k)), where C is some constant. Our result improves the previous best times O*((k/1.35)^k) for Line Cover (where d=2), O*(k^(dk)) for general Curve Cover, as well as a few other bounds for covering points by parabolas or conics. We also present an algorithm for Hyperplane Cover in R^3 with running time O*((Ck^2/log^(1/5) k)^k), improving on the previous time of O*((k^2/1.3)^k).
Cite as

Peyman Afshani, Edvin Berglin, Ingo van Duijn, and Jesper Sindahl Nielsen. Applications of Incidence Bounds in Point Covering Problems. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 60:1-60:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{afshani_et_al:LIPIcs.SoCG.2016.60,
  author =	{Afshani, Peyman and Berglin, Edvin and van Duijn, Ingo and Sindahl Nielsen, Jesper},
  title =	{{Applications of Incidence Bounds in Point Covering Problems}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{60:1--60:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{Fekete, S\'{a}ndor and Lubiw, Anna},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.60},
  URN =		{urn:nbn:de:0030-drops-59527},
  doi =		{10.4230/LIPIcs.SoCG.2016.60},
  annote =	{Keywords: Point Cover, Incidence Bounds, Inclusion Exclusion, Exponential Algorithm}
}
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