### Abstract

Let A and B be two sets of n points in the plane, and let M be a (one-to-one) matching between .4 and B. Let rain(M), max(M), and Σ(M) denote the length of the shortest edge, the length of the longest edge, and the sum of the lengths of the edges of M respectively. The uniform matching problem (also called the balanced assignment problem, or the fair matching problem) is to find M^{*} _{U}, a matching that minimizes max(M) - rain(M). A minimum deviation matching M^{*} _{D} is a matching that (1/n)Σ(M) – min(M). We present algorithms for computing M^{*} _{U} and M^{*} _{D} in roughly O(n^{10/3}) time. These algorithms are more efficient than the previous O(n^{4})-time algorithms of Martello and Toth [19] and Gupta and Punnen [11], who studied these problems for general bipartite graphs. We also consider the (non-bipartite version of the) bottleneck matching problem in higher dimensions. We extend the planar results of Chang et al. [4] and Su and Chang [22], and show that given a set A of 2n points in d-space, it is possible to compute a bottleneck matching of A in roughly O(n^{3/2}) time, for d ≤ 6, and in subquadratic time, for d > 6.

Original language | English (US) |
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Title of host publication | Algorithms and Computation - 7th International Symposium, ISAAC 1996, Proceedings |

Publisher | Springer Verlag |

Pages | 116-125 |

Number of pages | 10 |

Volume | 1178 |

ISBN (Print) | 3540620486, 9783540620488 |

State | Published - 1996 |

Externally published | Yes |

Event | 7th International Symposium on Algorithms and Computation, ISAAC 1996 - Osaka, Japan Duration: Dec 16 1996 → Dec 18 1996 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 1178 |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 7th International Symposium on Algorithms and Computation, ISAAC 1996 |
---|---|

Country | Japan |

City | Osaka |

Period | 12/16/96 → 12/18/96 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Algorithms and Computation - 7th International Symposium, ISAAC 1996, Proceedings*(Vol. 1178, pp. 116-125). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1178). Springer Verlag.

**Computing fair and bottleneck matchings in geometric graphs.** / Efrat, Alon; Katz, Matthew J.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Algorithms and Computation - 7th International Symposium, ISAAC 1996, Proceedings.*vol. 1178, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 1178, Springer Verlag, pp. 116-125, 7th International Symposium on Algorithms and Computation, ISAAC 1996, Osaka, Japan, 12/16/96.

}

TY - GEN

T1 - Computing fair and bottleneck matchings in geometric graphs

AU - Efrat, Alon

AU - Katz, Matthew J.

PY - 1996

Y1 - 1996

N2 - Let A and B be two sets of n points in the plane, and let M be a (one-to-one) matching between .4 and B. Let rain(M), max(M), and Σ(M) denote the length of the shortest edge, the length of the longest edge, and the sum of the lengths of the edges of M respectively. The uniform matching problem (also called the balanced assignment problem, or the fair matching problem) is to find M* U, a matching that minimizes max(M) - rain(M). A minimum deviation matching M* D is a matching that (1/n)Σ(M) – min(M). We present algorithms for computing M* U and M* D in roughly O(n10/3) time. These algorithms are more efficient than the previous O(n4)-time algorithms of Martello and Toth [19] and Gupta and Punnen [11], who studied these problems for general bipartite graphs. We also consider the (non-bipartite version of the) bottleneck matching problem in higher dimensions. We extend the planar results of Chang et al. [4] and Su and Chang [22], and show that given a set A of 2n points in d-space, it is possible to compute a bottleneck matching of A in roughly O(n3/2) time, for d ≤ 6, and in subquadratic time, for d > 6.

AB - Let A and B be two sets of n points in the plane, and let M be a (one-to-one) matching between .4 and B. Let rain(M), max(M), and Σ(M) denote the length of the shortest edge, the length of the longest edge, and the sum of the lengths of the edges of M respectively. The uniform matching problem (also called the balanced assignment problem, or the fair matching problem) is to find M* U, a matching that minimizes max(M) - rain(M). A minimum deviation matching M* D is a matching that (1/n)Σ(M) – min(M). We present algorithms for computing M* U and M* D in roughly O(n10/3) time. These algorithms are more efficient than the previous O(n4)-time algorithms of Martello and Toth [19] and Gupta and Punnen [11], who studied these problems for general bipartite graphs. We also consider the (non-bipartite version of the) bottleneck matching problem in higher dimensions. We extend the planar results of Chang et al. [4] and Su and Chang [22], and show that given a set A of 2n points in d-space, it is possible to compute a bottleneck matching of A in roughly O(n3/2) time, for d ≤ 6, and in subquadratic time, for d > 6.

UR - http://www.scopus.com/inward/record.url?scp=84990228138&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84990228138&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84990228138

SN - 3540620486

SN - 9783540620488

VL - 1178

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 116

EP - 125

BT - Algorithms and Computation - 7th International Symposium, ISAAC 1996, Proceedings

PB - Springer Verlag

ER -