Minimum-weight cycle covers and their approximability

  • Authors:

  • Bodo Manthey

  • Affiliations:

  • Saarland University, Computer Science, Postfach 151150, 66041 Saarbrücken, Germany

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2009

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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-@e 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.