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On Approximating Restricted Cycle Covers

  • Bodo Manthey
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3879)

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. A special case of L-cycle covers are k-cycle covers for k ∈ ℕ, where the length of each cycle must be at least k. The weight of a cycle cover of an edge-weighted graph is the sum of the weights of its edges.

We come close to settling the complexity and approximability of computing L-cycle covers. On the one hand, we show that for almost all L, computing L-cycle covers of maximum weight in directed and undirected graphs is APX-hard and NP-hard. Most of our hardness results hold even if the edge weights are restricted to zero and one. On the other hand, we show that the problem of computing L-cycle covers of maximum weight can be approximated with factor 2.5 for undirected graphs and with factor 3 in the case of directed graphs. Finally, we show that 4-cycle covers of maximum weight in graphs with edge weights zero and one can be computed in polynomial time.

As a by-product, we show that the problem of computing minimum vertex covers in λ-regular graphs is APX-complete for every λ≥3.

Keywords

Approximation Algorithm Directed Graph Undirected Graph Edge Weight Travel Salesman Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Bodo Manthey
    • 1
  1. 1.Institut für Theoretische InformatikUniversität zu LübeckLübeckGermany

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