An O(N log N) Algorithm for a Class of Matching Problems

The following class of matching problems is considered. The vertices of a complete und~res-This class of weighted graphs is applicable to scheduhng and optimal assignment problems. A maximum weighted (perfect) matching is found in O(n. log n) operations. 1. Introduction. The maximum matching problem has many applications in operations research. The first polynomial-time bounded algorithm for the maximum weighted matching problem is Edmonds' [2]. The most efficient algorithm for the maximum (cardinality) matching, known to the authors, is Even and Kariv's [3]. Gabow [4] has the most efficient algorithm for the weighted matching. In this paper we focus on a subclass of maximum weighted matching problems (see • ˜2 for a precise definition). Our study is motivated by the following two problems which are easily shown to belong to our class. In the first problem, a group of individuals, ordered by seniority, is to be partitioned into teams, having the same mission. Each team consists of two positions a senior position and a junior one. The senior position must be manned by the more senior individual between the members of the teant. Assuming that we know the effectiveness of each individual in both the senior and the junior positions, we wish to maximize the total effectiveness of the teams. The second problem is to schedule 2m jobs to m identical processors, two jobs to each processor, preserving the arrival ordering. The objective is to minimize the total flow time, or equivalently, the average waiting time of a job. Using a dynamic programming approach, these two models can be solved in 0 (m 2) time. In this paper we present an algorithm which solves the above problems in