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Binary Matrix Completion Under Diameter Constraints

Authors Tomohiro Koana , Vincent Froese , Rolf Niedermeier



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Author Details

Tomohiro Koana
  • Algorithmics and Computational Complexity, Faculty IV, Technische Universität Berlin, Germany
Vincent Froese
  • Algorithmics and Computational Complexity, Faculty IV, Technische Universität Berlin, Germany
Rolf Niedermeier
  • Algorithmics and Computational Complexity, Faculty IV, Technische Universität Berlin, Germany

Acknowledgements

We are grateful to Christian Komusiewicz for helpful feedback on an earlier version of this work and to Stefan Szeider for pointing us to the work on clustering incomplete data [Eduard Eiben et al., 2019]. We also thank Curtis Bright for mentioning the connection to the Hadamard matrix completion problem.

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Tomohiro Koana, Vincent Froese, and Rolf Niedermeier. Binary Matrix Completion Under Diameter Constraints. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 47:1-47:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.STACS.2021.47

Abstract

We thoroughly study a novel but basic combinatorial matrix completion problem: Given a binary incomplete matrix, fill in the missing entries so that the resulting matrix has a specified maximum diameter (that is, upper-bounding the maximum Hamming distance between any two rows of the completed matrix) as well as a specified minimum Hamming distance between any two of the matrix rows. This scenario is closely related to consensus string problems as well as to recently studied clustering problems on incomplete data. We obtain an almost complete picture concerning the complexity landscape (P vs NP) regarding the diameter constraints and regarding the number of missing entries per row of the incomplete matrix. We develop polynomial-time algorithms for maximum diameter three, which are based on Deza’s theorem [Discret. Math. 1973, J. Comb. Theory, Ser. B 1974] from extremal set theory. In this way, we also provide one of the rare links between sunflower techniques and stringology. On the negative side, we prove NP-hardness for diameter at least four. For the number of missing entries per row, we show polynomial-time solvability when there is only one missing entry and NP-hardness when there can be at least two missing entries. In general, our algorithms heavily rely on Deza’s theorem and the correspondingly identified sunflower structures pave the way towards solutions based on computing graph factors and solving 2-SAT instances.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Mathematics of computing → Discrete mathematics
Keywords
  • sunflowers
  • binary matrices
  • Hamming distance
  • stringology
  • consensus problems
  • complexity dichotomy
  • combinatorial algorithms
  • graph factors
  • 2-Sat
  • Hamming radius

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