Minimum-weight cycle covers and their approximability

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 non-constructive polynomial-time approximation algorithms that achieve constant approximation ratios for all sets L . On the other hand, we prove that the problem cannot be approximated with a factor of 2 − ε for certain sets L . For directed graphs, we devise non-constructive polynomial-time approximation algorithms that achieve approximation ratios of O ( n ) , where n is the number of vertices. This is asymptotically optimal: We show that the problem cannot be approximated with a factor of o ( n ) for certain sets L . 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.