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Line Cover and Related Problems

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

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Subject Classification

ACM Subject Classification
  • Theory of computation → Random projections and metric embeddings
  • Theory of computation → Computational geometry
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Point Line Cover
  • Projective Clustering
  • W-hardness
  • XP algorithm

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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 Get BibTex

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) https://doi.org/10.4230/LIPIcs.STACS.2026.13

Author Details

Matthias Bentert
  • TU Berlin, Germany
  • University of Bergen, Norway
Fedor V. Fomin
  • University of Bergen, Norway
Petr A. Golovach
  • University of Bergen, Norway
Souvik Saha
  • The Institute of Mathematical Sciences, HBNI, Chennai, India
Sanjay Seetharaman
  • The Institute of Mathematical Sciences, HBNI, Chennai, India
Anannya Upasana
  • The Institute of Mathematical Sciences, HBNI, Chennai, India

Funding

  • Bentert, Matthias: Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 819416) and the German Research Foundation (DFG, grant 1-5003036: SPP 2378 - ReNO-2).
  • Fomin, Fedor V.: Supported by the Research Council of Norway under BWCA project, reference 314528, and European Research Council (ERC) via grant NewPC, reference 101199930.
  • Golovach, Petr A.: Supported by the Research Council of Norway under BWCA (grant no. 314528) and Extreme-Algorithms (grant no 355137) projects.

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