Abstract
Given an undirected weighted graph \(G=(V,E)\) with nonnegative weight function obeying the triangle inequality, a set \(\{C_1,C_2,\ldots ,C_k\}\) of cycles is called a cycle cover if \(V \subseteq \bigcup _{i=1}^k V(C_i)\) and its cost is given by the maximum weight of the cycles. The Minimum Cycle Cover Problem aims to find a cycle cover of cost at most \(\lambda \) with the minimum number of cycles. An \(O(n^2)\) 24/5-approximation algorithm and an \(O(n^5)\) 14/3-approximation algorithm are given by Yu and Liu (Theor Comput Sci 654:45–58, 2016). However, the original proofs for approximation ratios are incomplete. In this paper we first present a corrected simplified analysis on the 24/5-approximation algorithm. Then we give a new \(O(n^3)\) approximation algorithm that achieves the same ratio 14/3 and has much simpler proof on the approximation ratio. Moreover, we derive an improved 32/7-approximation algorithm that runs in \(O(n^5)\).
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Acknowledgements
The authors are grateful to the anonymous referees for their helpful comments. This research is supported in part by the National Natural Science Foundation of China under grants numbers 11671135, 11701363 and the Fundamental Research Fund for the Central Universities under grant number 22220184028.
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Yu, W., Liu, Z., Bao, X. (2018). New Approximation Algorithms for the Minimum Cycle Cover Problem. In: Chen, J., Lu, P. (eds) Frontiers in Algorithmics. FAW 2018. Lecture Notes in Computer Science(), vol 10823. Springer, Cham. https://doi.org/10.1007/978-3-319-78455-7_7
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DOI: https://doi.org/10.1007/978-3-319-78455-7_7
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