Efficient Kernelization Algorithm for Bipartite Graph Matching

TL;DR

The paper tackles accelerating exact maximum bipartite matching by kernelizing the input with Karp-Sipser data reduction rules. It introduces MVM, a near-linear-time kernelization algorithm that employs a multi-vertex merging strategy, indirect set operations, and balanced processing, supported by a CSR-like data structure to preserve locality during vertex merges. The authors prove an overall time bound of $O(min(m \log n, n^2))$ and demonstrate strong empirical performance, outperforming KaSi variants and providing robust acceleration when combined with exact matching solvers. The work yields practical gains for large-scale graphs by significantly reducing problem size while maintaining efficient neighborhood traversals, making exact matching more scalable in real-world and synthetic settings.

Abstract

Finding the maximum matching in bipartite graphs is a fundamental graph operation widely used in various fields. To expedite the acquisition of the maximum matching, Karp and Sipser introduced two data reduction rules aimed at decreasing the input size. However, the KaSi algorithm, which implements the two data reduction rules, has several drawbacks: a high upper bound on time complexity and inefficient storage structure. The poor upper bound on time complexity makes the algorithm lack robustness when dealing with extreme cases, and the inefficient storage structure struggles to balance vertex merging and neighborhood traversal operations, leading to poor performance on real-life graphs. To address these issues, we introduced MVM, an algorithm incorporating three novel optimization strategies to implement the data reduction rules. Our theoretical analysis proves that the MVM algorithm, even when using data structures with the worst search efficiency, can still maintain near-linear time complexity, ensuring the algorithm's robustness. Additionally, we designed an innovative storage format that supports efficient vertex merging operations while preserving the locality of edge sets, thus ensuring the efficiency of neighborhood traversals in graph algorithms. Finally, we conduct evaluations on both real-life and synthetic graphs. Extensive experiments demonstrate the superiority of our method.

Figures (11)

• Figure 1: An example of Karp and Sipser's data reduction rule, where the black solid lines represent edges related to the matching, and the black double lines represent edges related to update operations. Circle vertices and square vertices denote the two types of vertices in the bipartite graph.
• Figure 2: An example of connecting the edge tables of the boundary vertices, corresponding to the vertex merging operation in Figure \ref{['fig:sigmod_kasi_rule']}. Each vertex in the vertex array contains five pointers: $vtx\_ptr\_start$, $vtx\_ptr\_end$, $vtx \_link\_next$, $vtx\_link\_cur$, and $vtx\_link\_last$. The first two pointers are related to the positions of the elements in the edge array pointed to by the current vertex, while the last three pointers are associated with other edge tables connected to the current vertex.
• Figure 3: An example of updating the edge tables for external vertices, corresponding to the vertex merging operation shown in Figure \ref{['fig:sigmod_kasi_rule']}. In the edge array, $.$s represent connections that are unaffected by the merge operation, corresponding to the dashed lines in Figure \ref{['fig:sigmod_kasi_rule']}.
• Figure 4: The runtime of MVM with different optimization strategies.
• Figure 5: The runtime of MVM and the variants of KaSi algorithm.