Our objective is to find a minimum-cost maximum matching, i.e., one for which the sum of the edge costs has minimal value. This is the weighted bipartite matching problem or, as it is sometimes called, the assignment problem. We report a new and very fast algorithm for an abstract special case of this problem. In both cases only linear space is required. As such we expect that it may be of practical value in several problem areas.
Many natural graphs satisfy the quasi-convexity condition.
Given two symbol strings over the same alphabet, we may imagine one to be red and the other blue and use our algorithms to compute string distances. In this formulation, the strings are embedded in the real line and multiple independent assignment problems are solved, one for each distinct alphabet symbol.