FIMP-HGA: A Novel Approach to Addressing the Partitioning Min-Max Weighted Matching Problem

TL;DR

This paper presents a novel approach, the Fast Iterative Match-Partition Hybrid Genetic Algorithm (FIMP-HGA), for addressing PMMWM, and proposes the KM-M algorithm, which reduces matching complexity through incremental adjustments, significantly enhancing runtime efficiency.

Abstract

The Partitioning Min-Max Weighted Matching (PMMWM) problem, being a practical NP-hard problem, integrates the task of partitioning the vertices of a bipartite graph into disjoint sets of limited size with the classical Maximum-Weight Perfect Matching (MPWM) problem. Initially introduced in 2015, the state-of-the-art method for addressing PMMWM is the MP$_{\text{LS}}$. In this paper, we present a novel approach, the Fast Iterative Match-Partition Hybrid Genetic Algorithm (FIMP-HGA), for addressing PMMWM. Similar to MP$_{\text{LS}}$, FIMP-HGA divides the solving into match and partition stages, iteratively refining the solution. In the match stage, we propose the KM-M algorithm, which reduces matching complexity through incremental adjustments, significantly enhancing runtime efficiency. For the partition stage, we introduce a Hybrid Genetic Algorithm (HGA) incorporating an elite strategy and design a Greedy Partition Crossover (GPX) operator alongside a Multilevel Local Search (MLS) to optimize individuals in the population. Population initialization employs various methods, including the multi-way Karmarkar-Karp (KK) algorithm, ensuring both quality and diversity. At each iteration, the bipartite graph is adjusted based on the current solution, aiming for continuous improvement. To conduct comprehensive experiments, we develop a new instance generation method compatible with existing approaches, resulting in four benchmark groups. Extensive experiments evaluate various algorithm modules, accurately assessing each module's impact on improvement. Evaluation results on our benchmarks demonstrate that the proposed FIMP-HGA significantly enhances solution quality compared to MP$_{\text{LS}}$, meanwhile reducing runtime by 3 to 20 times.

Figures (6)

• Figure 1: (a) A feasible PMMWM solution. (b) Improve the solution quality from (a) by modifying the matching scheme.
• Figure 2: An illustrate of the KM-M algorithm.
• Figure 3: An example of the GPX operator for a vertex set of 10 elements (A - J) and three partitions.
• Figure 4: Trend of $R_{Opt}$ (a) and $T_{Opt}$ (b) for our proposed algorithmscga on Benchmark-All with $n = 50$ to $500$. $R_{Opt}$ and $T_{Opt}$ all increase with $n$ for all algorithm.
• Figure 5: Trend of $R_{Opt}$ for our proposed algorithms on Benchmark-Con with $Con = 0$ to $100$. Except for FIMP-HGA, the $R_{Opt}$ of other schemes obviously decreases as $Con$ increases.

Theorems & Definitions (6)

• Definition 1: Alternating Path
• Definition 2: Augmenting Path
• Definition 3: Feasible Label
• Definition 4: Equivalence Subgraph
• Theorem 1
• proof