## Abstract.

Let *A* and *B* be two sets of *n* objects in *\reals*^{ d } , and let *Match* be a (one-to-one) matching between *A* and *B* . Let min(*Match* ), max(*Match* ), and *Σ(Match)* denote the length of the shortest edge, the length of the longest edge, and the sum of the lengths of the edges of *Match* , respectively. *Bottleneck matching*— a matching that minimizes max(*Match* )— is suggested as a convenient way for measuring the resemblance between *A* and *B* . Several algorithms for computing, as well as approximating, this resemblance are proposed. The running time of all the algorithms involving planar objects is roughly *O(n*^{ 1.5 }*)* . For instance, if the objects are points in the plane, the running time of the exact algorithm is *O(n*^{ 1.5 }log * n )* . A semidynamic data structure for answering containment problems for a set of congruent disks in the plane is developed. This data structure may be of independent interest.

Next, the problem of finding a translation of *B* that maximizes the resemblance to *A* under the bottleneck matching criterion is considered. When *A* and *B* are point-sets in the plane, an *O(n*^{ 5 }log * n)* -time algorithm for determining whether for some translated copy the resemblance gets below a given *ρ* is presented, thus improving the previous result of Alt, Mehlhorn, Wagener, and Welzl by a factor of almost *n* . This result is used to compute the smallest such *ρ* in time *O(n*^{ 5 }log ^{ 2 }* n )* , and an efficient approximation scheme for this problem is also given.

The *uniform matching* problem (also called the *balanced assignment* problem, or the *fair matching* problem) is to find *Match*^{*}_{U} , a matching that minimizes max *(Match)-min(Match)* . A *minimum deviation matching**Match*^{*}_{D} is a matching that minimizes *(1/n)Σ(Match) - min(Match)* . Algorithms for computing *Match*^{*}_{U} and *Match*^{*}_{D} in roughly *O(n*^{ 10/3 }*)* time are presented. These algorithms are more efficient than the previous *O(n*^{ 4 }*)* -time algorithms of Martello, Pulleyblank, Toth, and de Werra, and of Gupta and Punnen, who studied these problems for general bipartite graphs.

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