Abstract
In this paper, we consider the submodular multicut problem in trees with linear penalties (SMCLP(T) problem). In the SMCLP(T) problem, we are given a tree \(T=(V,E)\) with a submodular function \(c(\cdot ): 2^{E}\rightarrow \mathbb {R}_{\ge 0}\), a set of k distinct pairs of vertices \(P=\{(s_1,t_1),(s_2,t_2),\ldots ,(s_k,t_k)\}\) with non-negative penalty costs \(\pi _{j}\) for the pairs \((s_j,t_j)\in P\). The goal is to find a partial multicut \(M\subseteq E\) such that the total cost, consisting of the submodular cost of M and the penalty cost of the pairs not cut by M, is minimized. Let \(P_j\) be the unique path from \(s_j\) to \(t_j\) in the tree, where \(1\le j\le k\) and let m be the maximal length of all \(P_j\). Our main work is to present an m-approximation algorithm for the SMCLP(T) problem via the primal-dual method.
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References
Garg, N., Vazirani, V.V., Yannakakis, M.: Primal-dual approximation algorithms for integral flow and multicut in trees. Algorithmica 18, 3–20 (1997)
Garg, N., Vazirani, V.V., Yannakakis, M.: Approximate max-flow min-(multi) cut theorems and their applications. SIAM J. Comput. 25(2), 235–251 (2006)
Fujishige, S.: Submodular Functions and Optimization, 2nd edn. Elsevier, Amsterdam (2005)
Iwata, S., Nagano, K.: Submodular function minimization under covering constraints, In: The 50th annual symposium on foundations of computer science, FOCS, pp. 671–680 (2009)
Kamiyama, N.: A note on the submodular vertex cover problem with submodular penalties. Theor. Comput. Sci. 659, 95–97 (2017)
Levin, A., Segev, D.: Partial multicuts in trees. Theor. Comput. Sci. 369(1–3), 384–395 (2006)
Liu, X.F., Li, W.D.: A primal dual approximation algorithm for the multicut problem in trees with submodular penalties, In: International Conference on Algorithmic Applications in Management, pp. 203–211 (2019)
Lovász, L.: Submodular Functions and Convexity. In: Bachm, A., Grtschel, M., Korte, B. (eds.) Mathematical Programing the State of the Art, pp. 235–237. Springer, Berlin (1983)
Xu, D.C., Wang, F.M., Du, D.L., Wu, C.C.: Approximation algorithms for submodular vertex cover problems with linear/submodular penalties using primal-dual technique. Theor. Comput. Sci. 630, 117–125 (2016)
Acknowledgements
The authors would like to thank the referee for giving this paper a careful reading and many valuable comments and useful suggestions. This work was supported by the NSF of China (No. 11971146), the NSF of Hebei Province of China (No. A2019205089, No. A2019205092), Hebei Province Foundation for Returnees (CL201714) and Overseas Expertise Introduction Program of Hebei Auspices (25305008).
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Hou, C., Gao, S., Liu, W. et al. An approximation algorithm for the submodular multicut problem in trees with linear penalties. Optim Lett 15, 1105–1112 (2021). https://doi.org/10.1007/s11590-020-01665-1
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DOI: https://doi.org/10.1007/s11590-020-01665-1