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# Computing a Minimum-Cost k-Hop Steiner Tree in Tree-Like Metrics

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LIPIcs.MFCS.2020.18.pdf
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## Acknowledgements

We thank Jiří Sgall for discussions on k-hop MST on paths and an anonymous reviewer for pointing us to the connection between highway dimension and doubling dimension.

## Cite As

Martin Böhm, Ruben Hoeksma, Nicole Megow, Lukas Nölke, and Bertrand Simon. Computing a Minimum-Cost k-Hop Steiner Tree in Tree-Like Metrics. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 18:1-18:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.MFCS.2020.18

## Abstract

We consider the problem of computing a Steiner tree of minimum cost under a k-hop constraint which requires the depth of the tree to be at most k. Our main result is an exact algorithm for metrics induced by graphs of bounded treewidth that runs in time n^O(k). For the special case of a path, we give a simple algorithm that solves the problem in polynomial time, even if k is part of the input. The main result can be used to obtain, in quasi-polynomial time, a near-optimal solution that violates the k-hop constraint by at most one hop for more general metrics induced by graphs of bounded highway dimension and bounded doubling dimension.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Graph algorithms analysis
• Mathematics of computing → Combinatorial optimization
##### Keywords
• k-hop Steiner tree
• dynamic programming
• bounded treewidth

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## References

1. Ittai Abraham, Daniel Delling, Amos Fiat, Andrew V. Goldberg, and Renato F. Werneck. Highway dimension and provably efficient shortest path algorithms. J. ACM, 63(5):41:1-41:26, 2016.
2. Ittai Abraham, Amos Fiat, Andrew V Goldberg, and Renato F Werneck. Highway dimension, shortest paths, and provably efficient algorithms. In Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms, pages 782-793, 2010.
3. Laurent Alfandari and Vangelis Th. Paschos. Approximating minimum spanning tree of depth 2. International Transactions in Operational Research, 6(6):607-622, 1999.
4. Ernst Althaus, Stefan Funke, Sariel Har-Peled, Jochen Könemann, Edgar A. Ramos, and Martin Skutella. Approximating k-hop minimum-spanning trees. Oper. Res. Lett., 33(2):115-120, 2005.
5. Omer Angel, Abraham D. Flaxman, and David B. Wilson. A sharp threshold for minimum bounded-depth and bounded-diameter spanning trees and steiner trees in random networks. Combinatorica, 32(1):1-33, 2012.
6. Sunil Arya, Gautam Das, David M. Mount, Jeffrey S. Salowe, and Michiel Smid. Euclidean spanners: Short, thin, and lanky. In Proceedings of the Twenty-Seventh Annual ACM Symposium on Theory of Computing, page 489–498, 1995.
7. Anantaram Balakrishnan and Kemal Altinkemer. Using a hop-constrained model to generate alternative communication network design. INFORMS Journal on Computing, 4(2):192-205, 1992.
8. Judit Bar-Ilan, Guy Kortsarz, and David Peleg. Generalized submodular cover problems and applications. Theor. Comput. Sci., 250(1-2):179-200, 2001.
9. Marshall Bern and Paul Plassmann. The steiner problem with edge lengths 1 and 2. Information Processing Letters, 32(4), 1989.
10. Hans L. Bodlaender. A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput., 25(6):1305-1317, 1996.
11. Paz Carmi, Lilach Chaitman-Yerushalmi, and Ohad Trabelsi. Bounded-hop communication networks. Algorithmica, 80(11):3050-3077, 2018.
12. Markus Chimani, Petra Mutzel, and Bernd Zey. Improved steiner tree algorithms for bounded treewidth. Journal of Discrete Algorithms, 16:67-78, 2012.
13. Markus Chimani and Joachim Spoerhase. Network Design Problems with Bounded Distances via Shallow-Light Steiner Trees. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015), volume 30, pages 238-248, 2015.
14. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Treewidth. In Parameterized algorithms, chapter 7, pages 151-244. Springer, Cham, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3_7.
15. Geir Dahl, Luís Gouveia, and Cristina Requejo. On formulations and methods for the hop-constrained minimum spanning tree problem. In Mauricio G. C. Resende and Panos M. Pardalos, editors, Handbook of Optimization in Telecommunications, pages 493-515. Springer, Boston, MA, 2006.
16. Michael Elkin and Shay Solomon. Narrow-shallow-low-light trees with and without steiner points. SIAM Journal on Discrete Mathematics, 25(1):181-210, 2011.
17. Andreas Emil Feldmann, Wai Shing Fung, Jochen Könemann, and Ian Post. A (1+ε)-embedding of low highway dimension graphs into bounded treewidth graphs. SIAM Journal on Computing, 47(4):1667-1704, 2018.
18. Luis Gouveia. Using the miller-tucker-zemlin constraints to formulate a minimal spanning tree problem with hop constraints. Comput. Oper. Res., 22(9):959-970, 1995.
19. Luis Gouveia, Luidi Simonetti, and Eduardo Uchoa. Modeling hop-constrained and diameter-constrained minimum spanning tree problems as steiner tree problems over layered graphs. Math. Program., 128(1-2):123-148, 2011.
20. Sudipto Guha and Samir Khuller. Greedy strikes back: Improved facility location algorithms. Journal of Algorithms, 31(1):228-248, 1999.
21. Anupam Gupta, Robert Krauthgamer, and James R. Lee. Bounded geometries, fractals, and low-distortion embeddings. In FOCS, pages 534-543. IEEE Computer Society, 2003.
22. Martin Haenggi. Twelve reasons not to route over many short hops. VTC2004-Fall, 5:3130-3134 Vol. 5, 2004.
23. Mohammad Taghi Hajiaghayi, Guy Kortsarz, and Mohammad R. Salavatipour. Approximating buy-at-bulk and shallow-light k-steiner trees. Algorithmica, 53(1):89-103, 2009.
24. Vojtěch Jarník and Miloš Kössler. O minimálních grafech, obsahujících n daných bodů. Časopis pro pěstování matematiky a fysiky, 63(8):223-235, 1934.
25. Erez Kantor and David Peleg. Approximate hierarchical facility location and applications to the bounded depth steiner tree and range assignment problems. J. Discrete Algorithms, 7(3):341-362, 2009.
26. Bernhard Korte and Jaroslav Nešetřil. Vojtěch Jarník’s work in combinatorial optimization. Discrete Mathematics, 235(1-3):1-17, 2001.
27. Guy Kortsarz and David Peleg. Approximating shallow-light trees. In Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, page 103–110, 1997.
28. Sören Laue and Domagoj Matijević. Approximating k-hop minimum spanning trees in euclidean metrics. Inf. Process. Lett., 107(3-4):96-101, 2008.
29. Shi Li. A 1.488 approximation algorithm for the uncapacitated facility location problem. Information and Computation, 222:45-58, 2013.
30. Prabhu Manyem and Matthias FM Stallmann. Some approximation results in multicasting. Technical report, North Carolina State University at Raleigh, USA, 1996.
31. Vikram Raj Saksena. Topological analysis of packet networks. IEEE Journal on Selected Areas in Communications, 7(8):1243-1252, 1989.
32. Stefan Voß. The steiner tree problem with hop constraints. Annals OR, 86:321-345, 1999.
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