Document

# A Constant Factor Approximation for Capacitated Min-Max Tree Cover

## File

LIPIcs.APPROX-RANDOM.2020.55.pdf
• Filesize: 471 kB
• 13 pages

## Cite As

Syamantak Das, Lavina Jain, and Nikhil Kumar. A Constant Factor Approximation for Capacitated Min-Max Tree Cover. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 55:1-55:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.55

## Abstract

Given a graph G = (V,E) with non-negative real edge lengths and an integer parameter k, the (uncapacitated) Min-Max Tree Cover problem seeks to find a set of at most k trees which together span V and each tree is a subgraph of G. The objective is to minimize the maximum length among all the trees. In this paper, we consider a capacitated generalization of the above and give the first constant factor approximation algorithm. In the capacitated version, there is a hard uniform capacity (λ) on the number of vertices a tree can cover. Our result extends to the rooted version of the problem, where we are given a set of k root vertices, R and each of the covering trees is required to include a distinct vertex in R as the root. Prior to our work, the only result known was a (2k-1)-approximation algorithm for the special case when the total number of vertices in the graph is kλ [Guttmann-Beck and Hassin, J. of Algorithms, 1997]. Our technique circumvents the difficulty of using the minimum spanning tree of the graph as a lower bound, which is standard for the uncapacitated version of the problem [Even et al.,OR Letters 2004] [Khani et al.,Algorithmica 2010]. Instead, we use Steiner trees that cover λ vertices along with an iterative refinement procedure that ensures that the output trees have low cost and the vertices are well distributed among the trees.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Routing and network design problems
##### Keywords
• Approximation Algorithms
• Graph Algorithms
• Min-Max Tree Cover
• Vehicle Routing
• Steiner Tree

## Metrics

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

## References

1. Noga Alon, Yossi Azar, Gerhard J Woeginger, and Tal Yadid. Approximation schemes for scheduling on parallel machines. Journal of Scheduling, 1(1):55-66, 1998.
2. Hyung-Chan An, Aditya Bhaskara, Chandra Chekuri, Shalmoli Gupta, Vivek Madan, and Ola Svensson. Centrality of trees for capacitated \$\$k\$\$k-center. Math. Program., 154(1-2):29-53, December 2015. URL: https://doi.org/10.1007/s10107-014-0857-y.
3. Esther M. Arkin, Refael Hassin, and Asaf Levin. Approximations for minimum and min-max vehicle routing problems. J. Algorithms, 59(1):1-18, 2006. URL: https://doi.org/10.1016/j.jalgor.2005.01.007.
4. Amariah Becker, Philip N. Klein, and Aaron Schild. A PTAS for bounded-capacity vehicle routing in planar graphs. CoRR, abs/1901.07032, 2019. URL: http://arxiv.org/abs/1901.07032.
5. Amariah Becker and Alice Paul. A PTAS for minimum makespan vehicle routing in trees. CoRR, abs/1807.04308, 2018. URL: http://arxiv.org/abs/1807.04308.
6. Jaroslaw Byrka, Bartosz Rybicki, and Sumedha Uniyal. An approximation algorithm for uniform capacitated k-median problem with 1+backslashepsilon capacity violation. In Integer Programming and Combinatorial Optimization - 18th International Conference, IPCO 2016, Liège, Belgium, June 1-3, 2016, Proceedings, pages 262-274, 2016. URL: https://doi.org/10.1007/978-3-319-33461-5_22.
7. Lin Chen, Klaus Jansen, Wenchang Luo, and Guochuan Zhang. An efficient PTAS for parallel machine scheduling with capacity constraints. In Combinatorial Optimization and Applications - 10th International Conference, COCOA 2016, Hong Kong, China, December 16-18, 2016, Proceedings, pages 608-623, 2016. URL: https://doi.org/10.1007/978-3-319-48749-6_44.
8. Lin Chen and Dániel Marx. Covering a tree with rooted subtrees - parameterized and approximation algorithms. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 2801-2820, 2018. URL: https://doi.org/10.1137/1.9781611975031.178.
9. Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012.
10. Guy Even, Naveen Garg, Jochen Könemann, R. 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.
11. Naveen Garg. Saving an epsilon: A 2-approximation for the k-mst problem in graphs. In Proceedings of the Thirty-seventh Annual ACM Symposium on Theory of Computing, STOC '05, pages 396-402, New York, NY, USA, 2005. ACM. URL: https://doi.org/10.1145/1060590.1060650.
12. Nili Guttmann-Beck and Refael Hassin. Approximation algorithms for min–max tree partition. Journal of Algorithms, 24(2):266-286, 1997. URL: https://doi.org/10.1006/jagm.1996.0848.
13. Mordecai Haimovich and Alexander HG Rinnooy Kan. Bounds and heuristics for capacitated routing problems. Mathematics of operations Research, 10(4):527-542, 1985.
14. Mong-Jen Kao. Iterative partial rounding for vertex cover with hard capacities. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2638-2653. SIAM, 2017.
15. Jerome J Karaganis. On the cube of a graph. Canadian Mathematical Bulletin, 11(2):295-296, 1968.
16. 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, June 2014. URL: https://doi.org/10.1007/s00453-012-9740-5.
17. Hiroshi Nagamochi and Kohei Okada. Approximating the minmax rooted-tree cover in a tree. Information Processing Letters, 104(5):173-178, 2007. URL: https://doi.org/10.1016/j.ipl.2007.06.012.
18. Barna Saha and Aravind Srinivasan. A new approximation technique for resource-allocation problems. Random Struct. Algorithms, 52(4):680-715, 2018. URL: https://doi.org/10.1002/rsa.20756.
19. Sam Chiu-wai Wong. Tight algorithms for vertex cover with hard capacities on multigraphs and hypergraphs. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2626-2637. Society for Industrial and Applied Mathematics, 2017.
20. Zhou Xu and Qi Wen. Approximation hardness of min-max tree covers. Oper. Res. Lett., 38(3):169-173, May 2010. URL: https://doi.org/10.1016/j.orl.2010.02.004.
X

Feedback for Dagstuhl Publishing

### Thanks for your feedback!

Feedback submitted

### Could not send message

Please try again later or send an E-mail