A cycle cover of a graph is a spanning subgraph where each node is part of exactly one simple cycle. A k-cycle cover is a cycle cover where each cycle has length at least k. We call the decision problems whether a directed or undirected graph has a k-cycle cover k-DCC and k-UCC. Given a graph with edge weights one and two, Min-k-DCC and Min-k-UCC are the minimization problems of finding a k-cycle cover with minimum weight. We present factor 4/3 approximation algorithms for Min-k-DCC with running time O(n5/2) (independent of k). Specifically, we obtain a factor 4/3 approximation algorithm for the asymmetric travelling salesperson problem with distances one and two and a factor 2/3 approximation algorithm for the directed path packing problem with the same running time. On the other hand, we show that k-DCC is NP-complete for k ≥ 3 and that Min-k-DCC has no PTAS for k ≥ 4, unless P = NP. Furthermore, we design a polynomial time factor 7/6 approximation algorithm for Min-k-UCC. As a lower bound, we prove that Min-k-UCC has no PTAS for k ≥ 12, unless P = NP.
CITATION STYLE
Bläser, M., & Siebert, B. (2001). Computing cycle covers without short cycles. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 2161, pp. 368–379). Springer Verlag. https://doi.org/10.1007/3-540-44676-1_31
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