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# Approximating Directed Steiner Problems via Tree Embedding

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Bundit Laekhanukit. Approximating Directed Steiner Problems via Tree Embedding. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 74:1-74:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.ICALP.2016.74

## Abstract

Directed Steiner problems are fundamental problems in Combinatorial Optimization and Theoretical Computer Science. An important problem in this genre is the k-edge connected directed Steiner tree (k-DST) problem. In this problem, we are given a directed graph G on n vertices with edge-costs, a root vertex r, a set of h terminals T and an integer k. The goal is to find a min-cost subgraph H subseteq G that connects r to each terminal t in T by k edge-disjoint r, t-paths. This problem includes as special cases the well-known directed Steiner tree (DST) problem (the case k=1) and the group Steiner tree (GST) problem. Despite having been studied and mentioned many times in literature, e.g., by Feldman et al. [SODA'09, JCSS'12], by Cheriyan et al. [SODA'12, TALG'14], by Laekhanukit [SODA'14] and in a survey by Kortsarz and Nutov [Handbook of Approximation Algorithms and Metaheuristics], there was no known non-trivial approximation algorithm for k-DST for k >= 2 even in a special case that an input graph is directed acyclic and has a constant number of layers. If an input graph is not acyclic, the complexity status of k-DST is not known even for a very strict special case that k=2 and h=2. In this paper, we make a progress toward developing a non-trivial approximation algorithm for k-DST. We present an O(D*k^{D-1}*log(n))-approximation algorithm for k-DST on directed acyclic graphs (DAGs) with D layers, which can be extended to a special case of k-DST on "general graphs" when an instance has a D-shallow optimal solution, i.e., there exist k edge-disjoint r, t-paths, each of length at most D, for every terminal t in T. For the case k=1 (DST), our algorithm yields an approximation ratio of O(D*log(h)), thus implying an O(log^3(h))-approximation algorithm for DST that runs in quasi-polynomial-time (due to the height-reduction of Zelikovsky [Algorithmica'97]). Our algorithm is based on an LP-formulation that allows us to embed a solution to a tree-instance of GST, which does not preserve connectivity. We show, however, that one can randomly extract a solution of k-DST from the tree-instance of GST. Our algorithm is almost tight when k and D are constants since the case that k=1 and D=3 is NP-hard to approximate to within a factor of O(log(h)), and our algorithm archives the same approximation ratio for this special case. We also remark that the k^{1/4-epsilon}-hardness instance of k-DST is a DAG with 6 layers, and our algorithm gives O(k^5*log(n))-approximation for this special case. Consequently, as our algorithm works for general graphs, we obtain an O(D*k^{D-1}*log(n))-approximation algorithm for a D-shallow instance of the k edge-connected directed Steiner subgraph problem, where we wish to connect every pair of terminals by k edgedisjoint paths.
##### Keywords
• Approximation Algorithms
• Network Design
• Graph Connectivity
• Directed Graph

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## References

1. Parinya Chalermsook, Fabrizio Grandoni, and Bundit Laekhanukit. On survivable set connectivity. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 25-36, 2015. URL: http://dx.doi.org/10.1137/1.9781611973730.3.
2. Moses Charikar, Chandra Chekuri, To-Yat Cheung, Zuo Dai, Ashish Goel, Sudipto Guha, and Ming Li. Approximation algorithms for directed steiner problems. J. Algorithms, 33(1):73-91, 1999. URL: http://dx.doi.org/10.1006/jagm.1999.1042.
3. Joseph Cheriyan, Bundit Laekhanukit, Guyslain Naves, and Adrian Vetta. Approximating rooted steiner networks. ACM Transactions on Algorithms, 11(2):8:1-8:22, 2014. URL: http://dx.doi.org/10.1145/2650183.
4. Uriel Feige. A threshold of ln n for approximating set cover. J. ACM, 45(4):634-652, 1998. URL: http://dx.doi.org/10.1145/285055.285059.
5. Moran Feldman, Guy Kortsarz, and Zeev Nutov. Improved approximation algorithms for directed steiner forest. J. Comput. Syst. Sci., 78(1):279-292, 2012. URL: http://dx.doi.org/10.1016/j.jcss.2011.05.009.
6. Zachary Friggstad, Jochen Könemann, Young Kun-Ko, Anand Louis, Mohammad Shadravan, and Madhur Tulsiani. Linear programming hierarchies suffice for directed steiner tree. In Integer Programming and Combinatorial Optimization - 17th International Conference, IPCO 2014, Bonn, Germany, June 23-25, 2014. Proceedings, pages 285-296, 2014. URL: http://dx.doi.org/10.1007/978-3-319-07557-0_24.
7. Naveen Garg, Goran Konjevod, and R. Ravi. A polylogarithmic approximation algorithm for the group steiner tree problem. J. Algorithms, 37(1):66-84, 2000. URL: http://dx.doi.org/10.1006/jagm.2000.1096.
8. Christopher S. Helvig, Gabriel Robins, and Alexander Zelikovsky. An improved approximation scheme for the group steiner problem. Networks, 37(1):8-20, 2001. URL: http://dx.doi.org/10.1002/1097-0037(200101)37:1<8::AID-NET2>3.0.CO;2-R.
9. Guy Kortsarz and Zeev Nutov. Approximating minimum-cost connectivity problems. In Teofilo F. Gonzalez, editor, Handbook of Approximation Algorithms and Metaheuristics. Chapman and Hall/CRC, 2007. URL: http://dx.doi.org/10.1201/9781420010749.ch58.
10. Bundit Laekhanukit. Parameters of two-prover-one-round game and the hardness of connectivity problems. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 1626-1643, 2014. URL: http://dx.doi.org/10.1137/1.9781611973402.118.
11. Carsten Lund and Mihalis Yannakakis. On the hardness of approximating minimization problems. J. ACM, 41(5):960-981, 1994. URL: http://dx.doi.org/10.1145/185675.306789.
12. Zeev Nutov. Approximability status of survivable network problems. Preprint available at URL: http://www.openu.ac.il/home/nutov/Survivable-Network.pdf.
13. Thomas Rothvoß. Directed steiner tree and the lasserre hierarchy. CoRR, abs/1111.5473, 2011. URL: http://arxiv.org/abs/1111.5473.
14. Alexander Zelikovsky. A series of approximation algorithms for the acyclic directed steiner tree problem. Algorithmica, 18(1):99-110, 1997. URL: http://dx.doi.org/10.1007/BF02523690.