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 ⊆ ℕ.
We investigate how well L-cycle covers of minimum weight can be approximated. For undirected graphs, we devise a polynomial-time approximation algorithm that achieves a constant approximation ratio for all sets L. On the other hand, we prove that the problem cannot be approximated within a factor of 2 − ε for certain sets L.
For directed graphs, we present a polynomial-time approximation algorithm that achieves an approximation ratio of O(n), where n is the number of vertices. This is asymptotically optimal: We show that the problem cannot be approximated within a factor of o(n).
To contrast the results for cycle covers of minimum weight, we show that the problem of computing L-cycle covers of maximum weight can, at least in principle, be approximated arbitrarily well.
A full version of this work is available at http://arxiv.org/abs/cs/0609103.
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Manthey, B. (2007). Minimum-Weight Cycle Covers and Their Approximability. In: Brandstädt, A., Kratsch, D., Müller, H. (eds) Graph-Theoretic Concepts in Computer Science. WG 2007. Lecture Notes in Computer Science, vol 4769. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74839-7_18
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DOI: https://doi.org/10.1007/978-3-540-74839-7_18
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