Computing fair and bottleneck matchings in geometric graphs

Abstract

Let A and B be two sets of n points in the plane, and let M be a (one-to-one) matching between A and B. Let min(M), max(M), and ∑(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) − min(M). A minimum deviation matching M ^*_D is a matching that minimizes (1/n)∑(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 sub quadratic time, for d>6.