Algorithmica

, Volume 42, Issue 2, pp 121–139 | Cite as

Approximating Maximum Weight Cycle Covers in Directed Graphs with Weights Zero and One

Article

Abstract

A cycle cover of a graph is a spanning subgraph, each node of which is part of exactly one simple cycle. A k-cycle cover is a cycle cover where each cycle has length at least k. Given a complete directed graph with edge weights zero and one, Max-k-DDC(0,1) is the problem of finding a k-cycle cover with maximum weight. We present a 2/3 approximation algorithm for Max-k-DDC(0,1) with running time O(n 5/2). This algorithm yields a 4/3 approximation algorithm for Max-k-DDC(1,2) as well. Instances of the latter problem are complete directed graphs with edge weights one and two. The goal is to find a k-cycle cover with minimum weight. We particularly obtain a 2/3 approximation algorithm for the asymmetric maximum traveling salesman problem with distances zero and one and a 4/3 approximation algorithm for the asymmetric minimum traveling salesman problem with distances one and two. As a lower bound, we prove that Max-k-DDC(0,1) for k ≥ 3 and Max-k-UCC(0,1) (finding maximum weight cycle covers in undirected graphs) for k ≥ 7 are \APX-complete.

Combinatorial optimization Approximation algorithms Inapproximability Traveling salesman problem Cycle covers 

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Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Institut fur Theoretische Informatik, IFW B46.2, ETH Zurich, ETH Zentrum, 8092 ZurichSwitzerland
  2. 2.Institut fur Theoretische Informatik, Universitat zu Lubeck, Ratzeburger Allee 160, 23538 LubeckGermany

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