Document

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

## File

LIPIcs.ISAAC.2018.45.pdf
• Filesize: 439 kB
• 12 pages

## Cite As

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

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. J. Barát and D. Gerbner. Edge-Decomposition of Graphs into Copies of a Tree with Four Edges. The Electronic Journal of Combinatorics, 21(1):1-55, 2014.
2. C. Berge. Sur le couplage maximum d'un graphe. Comptes Rendus de l'Académie des Sciences, 247:258-259, 1958.
3. T. Dokka, M. Bougeret, V. Boudet, R. Giroudeau, and F.C.R. Spieksma. Approximation algorithms for the wafer to wafer integration problem. In Proceedings of the 10th International Workshop on Approximation and Online Algorithms (WAOA 2012), volume 7846 of LNCS, pages 286-297. Springer, 2013.
4. T. Dokka, Y. Crama, and F.C.R. Spieksma. Multi-dimensional vector assignment problems. Discrete Optimization, 14:111-125, 2014.
5. F. Dong, W. Yan, and F. Zhang. On the number of perfect matchings of line graphs. Discrete Applied Mathematics, 161(6):794-801, 2013.
6. 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. CoRR, abs/1807.01962, 2018. URL: http://arxiv.org/abs/1807.01962.
7. A. Figueroa, A. Goldstein, T. Jiang, M. Kurowski, A. Lingas, and M. Persson. Approximate clustering of fingerprint vectors with missing values. In Proceedings of the 2005 Australasian Symposium on Theory of Computing (CATS 2005), volume 41 of CRPIT, pages 57-60. Australian Computer Society, 2005.
8. D.S. Hochbaum and A. Levin. Covering the edges of bipartite graphs using K_2,2 graphs. Theoretical Computer Science, 411(1):1-9, 2010.
9. M. Jünger, G. Reinelt, and W.R. Pulleyblank. On partitioning the edges of graphs into connected subgraphs. Journal of Graph Theory, 9(4):539-549, 1985.
10. S. Onn and L.J. Schulman. The vector partition problem for convex objective functions. Mathematics of Operations Research, 26(3):583-590, 2001.
11. S. Reda, G. Smith, and L. Smith. Maximizing the functional yield of wafer-to-wafer 3-D integration. IEEE Transactions on Very Large Scale Integration (VLSI) Systems, 17(9):1357-1362, 2009.
12. C. Thomassen. Edge-decompositions of highly connected graphs into paths. In Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, volume 78, pages 17-26. Springer, 2008.
13. V.V. Vazirani. Approximation Algorithms. Springer-Verlag, Inc., New York, USA, 2001.
14. D.P. Williamson and D.B. Shmoys. The Design of Approximation Algorithms. Cambridge University Press, New York, USA, 1st edition, 2011.
X

Feedback for Dagstuhl Publishing