Abstract
We investigate the Clustered Steiner tree problem on metric graphs, which is a variant of the Steiner minimum tree problem. In this problem, the required vertices are partitioned into clusters, and the subtrees spanning different clusters must be disjoint in a feasible clustered Steiner tree. In this paper, it is shown that the problem is NP-hard even if the inter-cluster tree and all the local topologies are given, where a local topology specifies the tree structure of required vertices in the same cluster. We show that the Steiner ratio of this problem is lower and upper bounded by three and four, respectively. We also propose a \((\rho +2)\)-approximation algorithm, where \(\rho \) is the approximation ratio for the Steiner minimum tree problem, and the approximation ratio can be improved to \(\rho +1\) if the local topologies are given. Two variants of this problem are also studied. When the goal is to minimize the inter-cluster cost and ignore the cost of local trees, the problem can be solved in polynomial time. But it is NP-hard if we ask for the minimum cost of local trees among all solutions with minimum inter-cluster cost.
Similar content being viewed by others
References
Agrawal A, Klein P, Ravi R (1995) When trees collide: an approximation algorithm for the generalized Steiner problem on networks. SIAM J Comput 24(3):440–456
Bao X, Liu Z (2012) An improved approximation algorithm for the clustered traveling salesman problem. Inform Process Lett 112(23):908–910
Byrka J, Grandoni F, Rothvoß T, Sanità L (2010) An improved LP-based approximation for Steiner tree. Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC10ACM, New York, USA, pp 583–592
Chen YH, Lu CL, Tang CY (2003) On the full and bottleneck full Steiner tree problems. In: Warnow T, Zhu B (eds) Computing and Combinatorics, COCOON03, vol 2697., Lecture Notes in Computer ScienceSpringer, Berlin Heidelberg, pp 122–129
Cormen TH, Leiserson CE, Rivest RL, Stein C (2001) Introduction to algorithms, 2nd edn. The MIT Press, Cambridge
Ding W, Xue G (2014) Minimum diameter cost-constrained Steiner trees. J Comb Optim 27(1):32–48
Drake DE, Hougardy S (2004) On approximation algorithms for the terminal Steiner tree problem. Inform Process Lett 89:15–18
Garey MR, Graham RL, Johnson DS (1977) The complexity of computing Steiner minimal trees. SIAM J Appl Math 32(4):835–859
Garey MR, Johnson DS (1977) The rectilinear Steiner tree problem is NP-Complete. SIAM J Appl Math 32(4):826–834
Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-completeness. W. H. Freeman & Co, New York
Garg N, Konjevod G, Ravi R (1998) A polylogarithmic approximation algorithm for the group Steiner tree problem. Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA98SIAM, Philadelphia, PA, USA, pp 253–259
Guttmann-beck N, Hassin R, Khuller S, Raghavachari B (2000) Approximation algorithms with bounded performance guarantees for the clustered traveling salesman problem. Algorithmica 28:422–437
Hsieh SY, Yang SC (2007) Approximating the selected-internal Steiner tree. Theor Comput Sci 381(1–3): 288–291
Hsu TS, Tsai KH, Wang DW, Lee DT (2005) Two variations of the minimum Steiner problem. J Comb Optim 9(1):101–120
Huang CW, Lee CW, Gao HM, Hsieh SY (2013) The internal Steiner tree problem: hardness and approximations. J Complex 29(1):27–43
Karp R (1972) Reducibility among combinatorial problems. In: Miller R, Thatcher J (eds) Complexity of computer computations. Plenum Press, New York, pp 85–103
Li X, Zou F, Huang Y, Kim D, Wu W (2010) A better constant-factor approximation for selected-internal Steiner minimum tree. Algorithmica 56(3):333–341
Lin GH, Xue G (2002) On the terminal Steiner tree problem. Inform Process Lett 84(2):103–107
Lu CL, Tang CY, Lee RCT (2003) The full Steiner tree problem. Theor Comput Sci 306:55–67
Martinez F, de Pina J, Soares J (2007) Algorithms for terminal Steiner trees. Theor Comput Sci 389(1–2): 133–142
Robins G, Zelikovsky A (2005) Tighter bounds for graph Steiner tree approximation. SIAM J Discret Math 19:122–134
Wu BY, Chao KM (2004) Spanning trees and optimization problems. Chapman and Hall, Boca Raton
Zelikovsky A (1993) An 11/6-approximation algorithm for the network Steiner problem. Algorithmica 9(5):463–470
Zou F, Li X, Gao S, Wu W (2009) Node-weighted Steiner tree approximation in unit disk graphs. J Comb Optim 18(4):342–349
Acknowledgments
This work was supported in part by NSC 100-2221-E-194-036-MY3 and NSC 101-2221-E-194-025-MY3 from the National Science Council, Taiwan, R.O.C.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wu, B.Y., Lin, CW. On the clustered Steiner tree problem. J Comb Optim 30, 370–386 (2015). https://doi.org/10.1007/s10878-014-9772-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-014-9772-7