Abstract
A cycle cover of a graph is a set of cycles such that every vertex is part of exactly one cycle. An L-cycle cover is a cycle cover in which the length of every cycle is in the set L. A special case of L-cycle covers are k-cycle covers for k ∈ ℕ, where the length of each cycle must be at least k. The weight of a cycle cover of an edge-weighted graph is the sum of the weights of its edges.
We come close to settling the complexity and approximability of computing L-cycle covers. On the one hand, we show that for almost all L, computing L-cycle covers of maximum weight in directed and undirected graphs is APX-hard and NP-hard. Most of our hardness results hold even if the edge weights are restricted to zero and one. On the other hand, we show that the problem of computing L-cycle covers of maximum weight can be approximated with factor 2.5 for undirected graphs and with factor 3 in the case of directed graphs. Finally, we show that 4-cycle covers of maximum weight in graphs with edge weights zero and one can be computed in polynomial time.
As a by-product, we show that the problem of computing minimum vertex covers in λ-regular graphs is APX-complete for every λ≥3.
A full version of this work is available at http://arxiv.org/abs/cs/0504038.
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Manthey, B. (2006). On Approximating Restricted Cycle Covers. In: Erlebach, T., Persinao, G. (eds) Approximation and Online Algorithms. WAOA 2005. Lecture Notes in Computer Science, vol 3879. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11671411_22
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DOI: https://doi.org/10.1007/11671411_22
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