Authors’ response to “an addendum on the incremental assignment problem” by Volgenant

Abstract

In [3] we introduced the incremental assignment problem. In the incremental assignment problem a new pair of nodes, and their incident edges are added to a weighted bipartite graph whose maximum weighted matching is already known, and maximum weighted matching of the extended graph is sought. We have proposed an O(V ) algorithm for the problem. Neither of the two papers [1] and [2] identify the “Incremental Assignment Problem” of [3]. In both of these papers some improvements have been proposed to the solution of the standart k × k assignment problem by using incremental approaches. Paper [3], on the other hand, introduces this new problem, proposes an algorithm specific to this problem and studies the complexity and the correctness of the solution in detail. Even though the papers [1] and [2] uses some incremental approaches to solve standart assignment problem, they don’t study the complexity and the correctness of their algorithms’ incremental steps in detail either. In [1] the solution of standart n×n assignment problem was improved by using some sort of incremental approach. Although the idea of using incremental approach and extending the solutions from size k−1 to solution size k resembles the approach in [3], the details of incrementalization is completely different. In [1] there is a row-wise incrementalization, on the other hand, in [3] the matrix (k− 1)× (k− 1) is used to construct k× k matrix. The paper focuses on different improvements also, such as using shortest augmenting paths etc. and studies the performance of their improvements emprically.