Document

# A Fast 3-Approximation for the Capacitated Tree Cover Problem with Edge Loads

## File

LIPIcs.SWAT.2024.39.pdf
• Filesize: 0.69 MB
• 14 pages

## Cite As

Benjamin Rockel-Wolff. A Fast 3-Approximation for the Capacitated Tree Cover Problem with Edge Loads. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 39:1-39:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SWAT.2024.39

## Abstract

The capacitated tree cover problem with edge loads is a variant of the tree cover problem, where we are given facility opening costs, edge costs and loads, as well as vertex loads. We try to find a tree cover of minimum cost such that the total edge and vertex load of each tree does not exceed a given bound. We present an 𝒪(mlog n) time 3-approximation algorithm for this problem. This is achieved by starting with a certain LP formulation. We give a combinatorial algorithm that solves the LP optimally in time 𝒪(mlog n). Then, we show that a linear time rounding and splitting technique leads to an integral solution that costs at most 3 times as much as the LP solution. Finally, we prove that the integrality gap of the LP is 3, which shows that we can not improve the rounding step in general.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Approximation algorithms
• Mathematics of computing → Trees
##### Keywords
• Approximation Algorithms
• Tree Cover
• LP

## Metrics

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

## References

1. Esther M Arkin, Refael Hassin, and Asaf Levin. Approximations for minimum and min-max vehicle routing problems. Journal of Algorithms, 59(1):1-18, 2006. URL: https://doi.org/10.1016/J.JALGOR.2005.01.007.
2. Christoph Bartoschek. Fast Repeater Tree Construction. PhD thesis, Rheinische Friedrich-Wilhelms-Universität Bonn, 2014.
3. Guy Even, Naveen Garg, Jochen Könemann, Ramamoorthi Ravi, and Amitabh Sinha. Min-max tree covers of graphs. Operations Research Letters, 32(4):309-315, 2004. URL: https://doi.org/10.1016/J.ORL.2003.11.010.
4. Stephan Held, Bernhard Korte, Dieter Rautenbach, and Jens Vygen. Combinatorial optimization in vlsi design. Combinatorial Optimization - Methods and Applications, 31:33-96, 2011. URL: https://doi.org/10.3233/978-1-60750-718-5-33.
5. Dorit S Hochbaum and David B Shmoys. A best possible heuristic for the k-center problem. Mathematics of operations research, 10(2):180-184, 1985. URL: https://doi.org/10.1287/MOOR.10.2.180.
6. Tapas Kanungo, David M Mount, Nathan S Netanyahu, Christine D Piatko, Ruth Silverman, and Angela Y Wu. A local search approximation algorithm for k-means clustering. In Proceedings of the 18th annual symposium on Computational geometry, pages 10-18. ACM, 2002. URL: https://doi.org/10.1145/513400.513402.
7. M. Reza Khani and Mohammad R. Salavatipour. Improved approximation algorithms for the min-max tree cover and bounded tree cover problems. Algorithmica, 69(2):443-460, 2014. URL: https://doi.org/10.1007/S00453-012-9740-5.
8. Samir Khuller and Yoram J Sussmann. The capacitated k-center problem. SIAM Journal on Discrete Mathematics, 13(3):403-418, 2000. URL: https://doi.org/10.1137/S0895480197329776.
9. Stuart Lloyd. Least squares quantization in pcm. IEEE transactions on information theory, 28(2):129-137, 1982. URL: https://doi.org/10.1109/TIT.1982.1056489.
10. Jens Maßberg and Jens Vygen. Approximation algorithms for a facility location problem with service capacities. ACM Transactions of Algorithms, 4(4):50:1-50:15, 2008. URL: https://doi.org/10.1145/1383369.1383381.
11. Benjamin Rockel-Wolff. A fast 3-approximation for the capacitated tree cover problem with edge loads, 2024. URL: https://arxiv.org/abs/2404.10638.
12. Stephan Schwartz. An overview of graph covering and partitioning. Discrete Mathematics, 345(8):112884-112900, 2022. URL: https://doi.org/10.1016/J.DISC.2022.112884.
13. Vera Traub and Thorben Tröbst. A fast (2+2/7)-approximation algorithm for capacitated cycle covering. Mathematical Programming, 192(1):497-518, 2022. URL: https://doi.org/10.1007/S10107-021-01678-3.
14. Zhou Xu, Dongsheng Xu, and Wenbin Zhu. Approximation results for a min-max location-routing problem. Discrete Applied Mathematics, 160(3):306-320, 2012. URL: https://doi.org/10.1016/J.DAM.2011.09.014.
15. Wei Yu and Zhaohui Liu. Better approximability results for min-max tree/cycle/path cover problems. Journal of Combinatorial Optimization, 37(2):563-578, 2019. URL: https://doi.org/10.1007/S10878-018-0268-8.
X

Feedback for Dagstuhl Publishing