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Partitioning Vectors into Quadruples: Worst-Case Analysis of a Matching-Based Algorithm

Authors Annette M. C. Ficker, Thomas Erlebach , Matús Mihalák , Frits C. R. Spieksma

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  • Mathematics of computing → Approximation algorithms
Keywords
  • approximation algorithm
  • matching
  • clustering problem

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Abstract

Consider a problem where 4k given vectors need to be partitioned into k clusters of four vectors each. A cluster of four vectors is called a quad, and the cost of a quad is the sum of the component-wise maxima of the four vectors in the quad. The problem is to partition the given 4k vectors into k quads with minimum total cost. We analyze a straightforward matching-based algorithm and prove that this algorithm is a 3/2-approximation algorithm for this problem. We further analyze the performance of this algorithm on a hierarchy of special cases of the problem and prove that, in one particular case, the algorithm is a 5/4-approximation algorithm. Our analysis is tight in all cases except one.

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Annette M. C. Ficker, Thomas Erlebach, Matús Mihalák, and Frits C. R. Spieksma. Partitioning Vectors into Quadruples: Worst-Case Analysis of a Matching-Based Algorithm. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 45:1-45:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.ISAAC.2018.45

Author Details

Annette M. C. Ficker
  • Faculty of Economics and Business, KU Leuven, Leuven, Belgium
Thomas Erlebach
  • Department of Informatics, University of Leicester, Leicester, United Kingdom
Matús Mihalák
  • Department of Data Science and Knowledge Engineering, Maastricht University, Maastricht, The Netherlands
Frits C. R. Spieksma
  • Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands

Funding

  • Erlebach, Thomas: Supported by a study leave granted by University of Leicester.

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