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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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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

Cite AsGet BibTex

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

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.

Subject Classification

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

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