Computing Cycle Covers without Short Cycles

  • Markus Bläser
  • Bodo Siebert
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2161)


A cycle cover of a graph is a spanning subgraph where each node is part of exactly one simple cycle. A k-cycle cover is a cycle cover where each cycle has length at least k. We call the decision problems whether a directed or undirected graph has a k-cycle cover k-DCC and k-UCC. Given a graph with edge weights one and two, Min-k-DCC and Min-k-UCC are the minimization problems of finding a k-cycle cover with minimum weight.

We present factor 4/3 approximation algorithms for Min-k-DCC with running time O(n 5/2 ) (independent of k). Specifically, we obtain a factor 4/3 approximation algorithm for the asymmetric travelling salesperson problem with distances one and two and a factor 2/3 approximation algorithm for the directed path packing problem with the same running time. On the other hand, we show that k-DCC is \( \mathcal{N}\mathcal{P} \)-complete for k ≥ 3 and that Min-k-DCC has no PTAS for k ≥ 4, unless \( \mathcal{P} = \mathcal{N}\mathcal{P} \)

Furthermore, we design a polynomial time factor 7/6 approximation algorithm for Min-k-UCC. As a lower bound, we prove that Min-k-UCC has no PTAS for k = 12, unless \( \mathcal{P} = \mathcal{N}\mathcal{P} \).


Approximation Algorithm Polynomial Time Variable Node Span Subgraph Travel Salesperson 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 2001

Authors and Affiliations

  • Markus Bläser
    • 1
  • Bodo Siebert
    • 1
  1. 1.Institut für Theoretische InformatikMed. Universität zu LübeckLübeckGermany

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