EEMCS

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Official URL: http://dx.doi.org/10.1016/j.dam.2008.10.005

Abstract

A cycle cover of a graph is a set of cycles such that every vertex is part of exactly one cycle. An -cycle cover is a cycle cover in which the length of every cycle is in the set .
We investigate how well -cycle covers of minimum weight can be approximated. For undirected graphs, we devise non-constructive polynomial-time approximation algorithms that achieve constant approximation ratios for all sets . On the other hand, we prove that the problem cannot be approximated with a factor of for certain sets .
For directed graphs, we devise non-constructive polynomial-time approximation algorithms that achieve approximation ratios of , where is the number of vertices. This is asymptotically optimal: We show that the problem cannot be approximated with a factor of for certain sets .
To contrast the results for cycle covers of minimum weight, we show that the problem of computing -cycle covers of maximum weight can, at least in principle, be approximated arbitrarily well.

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