An Efficient Algorithm for the Partitioning Min-Max Weighted Matching Problem

Abstract: The Partitioning Min-Max Weighted Matching (PMMWM) problem is an NP-hard problem that combines the problem of partitioning a group of vertices of a bipartite graph into disjoint subsets with limited size and the classical Min-Max Weighted Matching (MMWM) problem. Kress et al. proposed this problem in 2015 and they also provided several algorithms, among which MP${\text{LS}}$ is the state-of-the-art. In this work, we observe there is a time bottleneck in the matching phase of MP${\text{LS}}$. Hence, we optimize the redundant operations during the matching iterations, and propose an efficient algorithm called the MP${\text{KM-M}}$ that greatly speeds up MP${\text{LS}}$. The bottleneck time complexity is optimized from $O(n^3)$ to $O(n^2)$. We also prove the correctness of MP${\text{KM-M}}$ by the primal-dual method. To test the performance on diverse instances, we generate various types and sizes of benchmarks, and carried out an extensive computational study on the performance of MP${\text{KM-M}}$ and MP${\text{LS}}$. The evaluation results show that our MP${\text{KM-M}}$ greatly shortens the runtime as compared with MP$_{\text{LS}}$ while yielding the same solution quality.