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)

Abstract

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(n5/2) (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 \( \mathcal{P} = \mathcal{N}\mathcal{P} \).

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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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