Geometry Helps in Bottleneck Matching and Related Problems

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 ^1.5log  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 ⁵log  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 ⁵log 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.