Abstract
In this paper, we consider the k-prize-collecting Steiner tree problem (k-PCST), extending both the prize-collecting Steiner tree problem (PCST) and the k-minimum spanning tree problem (k-MST). In this problem, we are given a connected graph \(G = (V, E)\), a root vertex r and an integer k. Every edge in E has a nonnegative cost. Every vertex in V has a nonnegative penalty cost. We want to find an r-rooted tree F that spans at least k vertices such that the total cost, including the edge cost of the tree F and the penalty cost of the vertices not spanned by F, is minimized. Our main contribution is to present a 5-approximation algorithm for the k-PCST via the methods of primal–dual and Lagrangean relaxation.
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Archer, A., Bateni, M.H., Hajiaghayi, M.T., Karloff, H.: Improved approximation algorithms for prize-collecting Steiner tree and TSP. SIAM J. Comput. 40, 309–332 (2011)
Arora, S, Karakostas, G.: A \(2+\varepsilon \) approximation algorithm for the \(k\)-MST problem. In: Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 754–759 (2000)
Arya, S., Ramesh, H.: A, \(2.5\)-factor approximation algorithm for the \(k\)-MST problem. Inf. Process. Lett. 65, 117–118 (1998)
Awerbuch, B., Azar, Y., Blum, A., Vempala, S.: Improved approximation guarantees for minimum-weight \(k\)-trees and prize-collecting salesmen. SIAM J. Comput. 28, 254–262 (1999)
Bienstock, D., Goemans, M.X., Simchi-Levi, D., Williamson, D.P.: A note on the prize collecting traveling salesman problem. Math. Programm. 59, 413–420 (1993)
Blum, A., Ravi, R., Vempala, S.: A constant-factor approximation algorithm for the \(k\)-MST problem. In: Proceedings of the 28th Annual ACM Symposium on Theory of Computing, pp. 442–448 (1996)
Chudak, F.A., Roughgarden, T., Williamson, D.P.: Approximate \(k\)-MSTs and \(k\)-Steiner trees via the primal–dual method and Lagrangean relaxation. Math. Programm. 100, 411–421 (2004)
M, Fischetti, HW, Hamacher, Jørnsten, K., Maffioli, F.: Weighted \(k\)-cardinality trees complexity and polyhedral structure. Networks 24, 11–21 (1994)
Garg, N.: A \(3\)-approximation for the minimum tree spanning \(k\) vertices. In: Proceedings of the 37th Annual Symposium on Foundations of Computer Science, pp. 302–309 (1996)
Garg, N.: Saving an epsilon: a \(2\)-approximation for the \(k\)-MST problem in graphs. In: Proceedings of the 37th Annual ACM Symposium on Theory of Computing, pp. 396–402 (2005)
Goemans, M.X., Williamson, D.P.: A general approximation technique for constrained forest problems. SIAM J. Comput. 24, 296–317 (1995)
Jain, K., Vazirani, V.V.: Approximation algorithms for metric facility location and \(k\)-median problems using the primal–dual schema and Lagrangian relaxation. J. ACM 48, 274–296 (2001)
Rajagopalan, S., Vazirani, V.V.: Logarithmic approximation of minimum weight \(k\) trees. Unpublished manuscript (1995)
Ravi, R., Sundaram, R., Marathe, M.V., Rosenkrantz, D.J., Ravi, S.S.: Spanning trees short or small. SIAM J. Discrete Math. 9, 178–200 (1996)
Acknowledgements
The second author is supported by Natural Science Foundation of China (No. 11371001) and Collaborative Innovation Center on Beijing Society-Building and Social Governance. The third author is supported by Natural Sciences and Engineering Research Council of Canada (NSERC) grant 06446 and Natural Science Foundation of China (No. 11271009). The fourth author is supported by Natural Science Foundation of China (No. 11501412).
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Han, L., Xu, D., Du, D. et al. A 5-approximation algorithm for the k-prize-collecting Steiner tree problem. Optim Lett 13, 573–585 (2019). https://doi.org/10.1007/s11590-017-1135-8
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DOI: https://doi.org/10.1007/s11590-017-1135-8