Abstract
In this paper, we introduce the submodular multicut problem in trees with submodular penalties, which generalizes the prize-collecting multicut problem in trees and the submodular vertex cover with submodular penalties. We present a combinatorial approximation algorithm, based on the primal-dual algorithm for the submodular set cover problem. In addition, we present a combinatorial 3-approximation algorithm for a special case where the edge cost is a modular function, based on the primal-dual scheme for the multicut problem in trees.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Dahlhaus E, Johnson DS, Papadimitriou CH, Seymour PD, Yannakakis M (1994) The complexity of multiterminal cuts. SIAM J Comput 23(4):864–894
Du D, Lu R, Xu D (2012) A primal-dual approximation algorithm for the facility location problem with submodular penalties. Algorithmica 63:191–200
Fujishige S (2005) Submodular functions and optimization, 2nd edn. Elsevier, Amsterdam
Garg N, Vazirani VV, Yannakakis M (1997) Primal-dual approximation algorithms for integral flow and multicut in trees. Algorithmica 18(1):3–20
Garg N, Vazirani VV, Yannakakis M (2006) Approximate max-flow min-(multi) cut theorems and their applications. SIAM J Comput 25(2):235–251
Hayrapetyan A, Swamy C, Tardos E (2005) Network design for information networks. In: Proceedings of the sixteenth annual ACM-SIAM symposium on discrete algorithms, pp 933–942
Hu TC (1969) Integer programming and network flows. Addison-Wesley, Reading, MA
Iwata S, Nagano K (2009) Submodular function minimization under covering constraints. In: The 50th annual symposium on foundations of computer science, FOCS, pp. 671–680
Iwata S, Fleischer L, Fujishige S (2001) A combinatorial strongly polynomial algorithm for minimizing submodular functions. J ACM 48(4):761–777
Kamiyama N (2017) A note on the submodular vertex cover problem with submodular penalties. Theor Comput Sci 659:95–97
Kanj I, Lin G, Liu T, Tong W, Xia G, Xu J, Yang B, Zhang F, Zhang P, Zhu B (2015) Improved parameterized and exact algorithms for cut problems on trees. Theor Comput Sci 607:455–470
Khot S, Regev O (2008) Vertex cover might be hard to approximateto with \(2-\epsilon \). J Comput Syst Sci 74(3):335–349
Levin A, Segev D (2006) Partial multicuts in trees. Theor Comput Sci 369(1–3):384–395
Li Y, Du D, Xiu N, Xu D (2015) Improved approximation algorithms for the facility location problems with linear/submodular penalties. Algorithmica 73(2):460–482
Liu H, Zhang P (2014) On the generalized multiway cut in trees problem. J Comb Optim 27(1):65–77
Xu D, Wang F, Du D, Wu C (2016) Approximation algorithms for submodular vertex cover problems with linear/submodular penalties using primal-dual technique. Theor Comput Sci 630:117–125
Zhang P, Zhu D, Luan J (2012) An approximation algorithm for the generalized k-multicut problem. Discrete Appl Math 160(7–8):1240–1247
Acknowledgements
The work is supported in part by the National Natural Science Foundation of China [No. 61662088], Program for Excellent Young Talents of Yunnan University, Training Program of National Science Fund for Distinguished Young Scholars, Project for Innovation Team (Cultivation) of Yunnan Province, IRTSTYN, and Key Joint Project of the Science and Technology Department of Yunnan Province and Yunnan University [No. 2018FY001(-014)].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Liu, X., Li, W. Combinatorial approximation algorithms for the submodular multicut problem in trees with submodular penalties. J Comb Optim 44, 1964–1976 (2022). https://doi.org/10.1007/s10878-020-00568-2
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-020-00568-2