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Low-Rank Binary Matrix Approximation in Column-Sum Norm

Authors Fedor V. Fomin , Petr A. Golovach , Fahad Panolan , Kirill Simonov

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  • Theory of computation → Approximation algorithms analysis
Keywords
  • Binary Matrix Factorization
  • PTAS
  • Column-sum norm

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Abstract

We consider 𝓁₁-Rank-r Approximation over {GF}(2), where for a binary m× n matrix 𝐀 and a positive integer constant r, one seeks a binary matrix 𝐁 of rank at most r, minimizing the column-sum norm ‖ 𝐀 -𝐁‖₁. We show that for every ε ∈ (0, 1), there is a {randomized} (1+ε)-approximation algorithm for 𝓁₁-Rank-r Approximation over {GF}(2) of running time m^{O(1)}n^{O(2^{4r}⋅ ε^{-4})}. This is the first polynomial time approximation scheme (PTAS) for this problem.

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Fedor V. Fomin, Petr A. Golovach, Fahad Panolan, and Kirill Simonov. Low-Rank Binary Matrix Approximation in Column-Sum Norm. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 32:1-32:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.32

Author Details

Fedor V. Fomin
  • Department of Informatics, University of Bergen, Norway
Petr A. Golovach
  • Department of Informatics, University of Bergen, Norway
Fahad Panolan
  • Department of Computer Science and Engineering, IIT Hyderabad, India
Kirill Simonov
  • Department of Informatics, University of Bergen, Norway

Funding

  • Fomin, Fedor V.: The research leading to these results have been supported by the Research Council of Norway via the project "MULTIVAL" (grant 263317). The work was done within the CEDAS center in Bergen.

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