2006/09/18 by Manthey, Bodo
#Computational Complexity (cs.CC) #Data Structures and Algorithms (cs.DS) #Discrete Mathematics (cs.DM) #F.2.2 #FOS: Computer and information sciences #G.2.1 #G.2.2
paper · doi:10.48550/arxiv.cs/0609103
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. 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-eps 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.