## Abstract

A directed multigraph is said to be d-regular if the indegree and outdegree of every vertex is exactly d. By Hall's theorem one can represent such a multigraph as a combination of at most n^{2} cycle covers each taken with an appropriate multiplicity. We prove that if the d-regular multigraph does not contain more than ⌊d/2⌋ copies of any 2-cycle then we can find a similar decomposition into 0(n^{2}) pairs of cycle covers where each 2-cycle occurs in at most one component of each pair. Our proof is constructive and gives a polynomial algorithm to find such decomposition. Since our applications only need one such a pair of cycle covers whose weight is at least the average weight of all pairs, we also give a simpler algorithm to extract a single such pair. This combinatorial theorem then comes handy in rounding a fractional solution of an LP relaxation of the maximum and minimum TSP problems. For maximum TSP, we obtain a tour whose weight is at least 2/3 of the weight of the longest tour, improving a previous 5/8 approximation. For minimum TSP we obtain a tour whose weight is at most 0.842log_{2} n times the optimal, improving a previous 0.999log_{2} n approximation. Utilizing a reduction from maximum TSP to the shortest superstring problem we obtain a 2.5-approximation algorithm for the latter problem which is again much simpler than the previous one. Other applications of the rounding procedure are approximation algorithms for maximum 3-cycle cover (factor 2/3, previously 3/5) and maximum asymmetric TSP with triangle inequality (factor 10/13, previously 3/4 ).

Original language | English |
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Title of host publication | Proceedings - 44th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2003 |

Publisher | IEEE Computer Society |

Pages | 56-65 |

Number of pages | 10 |

ISBN (Electronic) | 0769520405 |

DOIs | |

State | Published - 2003 |

Event | 44th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2003 - Cambridge, United States Duration: 11 Oct 2003 → 14 Oct 2003 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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Volume | 2003-January |

ISSN (Print) | 0272-5428 |

### Conference

Conference | 44th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2003 |
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Country/Territory | United States |

City | Cambridge |

Period | 11/10/03 → 14/10/03 |

### Bibliographical note

Publisher Copyright:© 2003 IEEE.

### Funding

Funders | Funder number |
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Israel Science Foundation | 548/00 |

## Keywords

- Algorithm design and analysis
- Application software
- Approximation algorithms
- Biology computing
- Computational biology
- Computer applications
- Computer science
- Polynomials
- Traveling salesman problems