Engineering Hypergraph $b$-Matching Algorithms

TL;DR

This work tackles the weighted hypergraph $b$-matching problem by engineering a scalable pipeline that combines exact data reductions, a greedy initial-solution heuristic, and a robust iterated local-search framework. The reductions prune and reformulate instances, often enabling exact solvers to operate on dramatically smaller cores, while the greedy and local-search components produce high-quality solutions quickly. Extensive experiments on real-world hypergraphs demonstrate practical effectiveness, with significant speedups and notable solution-quality improvements over existing approaches, and the method extends to standard graph matching as well. The results suggest strong practical impact for large-scale combinatorial optimization tasks in hypergraphs, including clustering, resource allocation, and scheduling, with an open-source release planned for broader adoption.

Abstract

Recently, researchers have extended the concept of matchings to the more general problem of finding $b$-matchings in hypergraphs broadening the scope of potential applications and challenges. The concept of $b$-matchings, where $b$ is a function that assigns positive integers to the vertices of the graph, is a natural extension of matchings in graphs, where each vertex $v$ is allowed to be matched to up to $b(v)$ edges, rather than just one. The weighted $b$-matching problem then seeks to select a subset of the hyperedges that fulfills the constraint and maximizes the weight. In this work, we engineer novel algorithms for this generalized problem. More precisely, we introduce exact data reductions for the problem as well as a novel greedy initial solution and local search algorithms. These data reductions allow us to significantly shrink the input size. This is done by either determining if a hyperedge is guaranteed to be in an optimum $b$-matching and thus can be added to our solution or if it can be safely ignored. Our iterated local search algorithm provides a framework for finding suitable improvement swaps of edges. Experiments on a wide range of real-world hypergraphs show that our new set of data reductions are highly practical, and our initial solutions are competitive for graphs and hypergraphs as well.

Figures (12)

• Figure 1: Examples for the first two reductions. Neighborhood Removal (NR): The weight of the orange edge dominates the sum of the $b(v)$-th heaviest weights per vertex (white), in this case the blue ones($10\geq 8$). There exist an optimal solution that contains the orange edge. Weighted Isolated Edge Removal (WIER): All edges form a clique and have a common vertex with capacity 1 each. The orange edge is part of an optimal matching, because it has the highest weight of the clique.
• Figure 2: Examples for three reductions. Weighted Edge Folding (WEF): The orange edge has exactly two non-adjacent neighbors (blue), that it dominates one by one, but not in total. The three edges can be folded and later be decided on. Weighted Twin (WT): The orange edges have exactly two independent non-adjacent neighbors (blue), that they dominate one by one, but not in total. The four edges can be folded and later be decided on. The Weighted Edge Folding is directly applied. Weighted Domination (WD): The orange edge is a subset of the blue edge, has a higher weight and they share a common vertex with capacity $1$.
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• Figure 5: Relative running time and effect of reductions.