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

TL;DR

This paper introduces an optimized algorithm for the NP-hard Partitioning Min-Max Weighted Matching problem, significantly reducing computation time while maintaining solution quality through improved matching phase efficiency.

Contribution

The authors propose the MP_KM-M algorithm, which optimizes the matching phase of the existing MP_LS algorithm, reducing time complexity from O(n^3) to O(n^2) and proving its correctness.

Findings

MP_KM-M greatly shortens runtime compared to MP_LS.

Solution quality remains consistent between MP_KM-M and MP_LS.

Extensive tests on diverse benchmarks validate the efficiency improvements.

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.