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# Tighter Approximation for the Uniform Cost-Distance Steiner Tree Problem

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## Cite As

Josefine Foos, Stephan Held, and Yannik Kyle Dustin Spitzley. Tighter Approximation for the Uniform Cost-Distance Steiner Tree Problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.19

## Abstract

Uniform cost-distance Steiner trees minimize the sum of the total length and weighted path lengths from a dedicated root to the other terminals. They are applied when the tree is intended for signal transmission, e.g. in chip design or telecommunication networks. They are a special case of general cost-distance Steiner trees, where different distance functions are used for total length and path lengths. We improve the best published approximation factor for the uniform cost-distance Steiner tree problem from 2.39 [Khazraei and Held, 2021] to 2.05. If we can approximate the minimum-length Steiner tree problem arbitrarily well, our algorithm achieves an approximation factor arbitrarily close to 1+1/√2. This bound is tight in the following sense. We also prove the gap 1+1/√2 between optimum solutions and the lower bound which we and all previous approximation algorithms for this problem use. Similarly to previous approaches, we start with an approximate minimum-length Steiner tree and split it into subtrees that are later re-connected. To improve the approximation factor, we split it into components more carefully, taking the cost structure into account, and we significantly enhance the analysis.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Approximation algorithms analysis
• Theory of computation → Routing and network design problems
##### Keywords
• cost-distance Steiner tree
• approximation algorithm
• uniform

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## References

1. Sanjeev Arora. Polynomial time approximation schemes for euclidean traveling salesman and other geometric problems. Journal of the ACM, 45(5):753-782, 1998.
2. Marcelo P.L. Benedito, Lehilton L.C. Pedrosa, and Hugo K.K. Rosado. On the inapproximability of the cable-trench problem. Procedia Computer Science, 195:39-48, 2021.
3. Jarosław Byrka, Fabrizio Grandoni, Thomas Rothvoss, and Laura Sanità. Steiner tree approximation via iterative randomized rounding. Journal of the ACM, 60(1):article 6, 2013.
4. Chandra Chekuri, Sanjeev Khanna, and Joseph Naor. A deterministic algorithm for the cost-distance problem. In Proc. ACM-SIAM symposium on Discrete Algorithms (SODA '01), 2001, pages 232-233. SIAM, 2001.
5. Miroslav Chlebík and Janka Chlebíková. The steiner tree problem on graphs: Inapproximability results. Theoretical Computer Science, 406(3):207-214, 2008.
6. Julia Chuzhoy, Anupam Gupta, Joseph Naor, and Amitabh Sinha. On the approximability of some network design problems. ACM Transactions on Algorithms, 4(2):article 23, 2008.
7. Siad Daboul, Stephan Held, Bento Natura, and Daniel Rotter. Global interconnect optimization. ACM Transactions on Design Automation of Electronic Systems, 2023. (to appear, conference paper in Proc. ICCAD '19). URL: https://doi.org/10.1145/3587044.
8. Fabrizio Grandoni and Thomas Rothvoß. Network design via core detouring for problems without a core. In International Colloquium on Automata, Languages, and Programming, pages 490-502, 2010.
9. Sudipto Guha, Adam Meyerson, and Kamesh Munagala. A constant factor approximation for the single sink edge installation problem. SIAM Journal on Computing, 38(6):2426-2442, 2009.
10. Longkun Guo, Nianchen Zou, and Yidong Li. Approximating the shallow-light steiner tree problem when cost and delay are linearly dependent. In Proc. International Symposium on Parallel Architectures, Algorithms and Programming, pages 99-103, 2014.
11. Stephan Held, Dirk Müller, Daniel Rotter, Rudolf Scheifele, Vera Traub, and Jens Vygen. Global routing with timing constraints. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 37(2):406-419, 2018.
12. Stephan Held and Daniel Rotter. Shallow-light steiner arborescences with vertex delays. In Proc. International Conference on Integer Programming and Combinatorial Optimization (IPCO '13), pages 229-241, 2013.
13. Stephan Held and Yannik.K.D. Spitzley. Further improvements on approximating the uniform cost-distance steiner tree problem. Technical report, Research Institute for Discrete Mathematics, University of Bonn, 2022. URL: https://arxiv.org/abs/2211.03830.
14. Raja Jothi and Balaji Raghavachari. Improved approximation algorithms for the single-sink buy-at-bulk network design problems. Journal of Discrete Algorithms, 7(2):249-255, 2009.
15. Ardalan Khazraei and Stephan Held. An improved approximation algorithm for the uniform cost-distance Steiner tree problem. In Proc. Workshop on Approximation and Online Algorithms (WAOA 2020), pages 189-203. Springer, 2021.
16. Samir Khuller, Balaji Raghavachari, and Neal E. Young. Balancing minimum spanning trees and shortest-path trees. Algorithmica, 14(4):305-321, 1995.
17. Adam Meyerson, Kamesh Munagala, and Serge Plotkin. Cost-distance: Two metric network design. SIAM Journal on Computing, 38(4):1648-1659, 2008.
18. Daniel Rotter. Timing-constrained global routing with buffered Steiner trees. PhD thesis, Universitäts-und Landesbibliothek Bonn, 2017.
19. Kunal Talwar. The single-sink buy-at-bulk lp has constant integrality gap. In International Conference on Integer Programming and Combinatorial Optimization, pages 475-486, 2002.
20. Vera Traub and Rico Zenklusen. Local search for weighted tree augmentation and steiner tree. In Proc. ACM-SIAM Symposium on Discrete Algorithms (SODA '22), pages 3253-3272, 2022.
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