Engineering Hypergraph $b$-Matching Algorithms

TL;DR

This paper develops new algorithms for the complex problem of finding maximum weighted $b$-matchings in hypergraphs, including data reductions, greedy initial solutions, and local search, demonstrating practical efficiency on real-world data.

Contribution

The work introduces novel exact data reductions, a greedy initial solution, and local search algorithms specifically designed for hypergraph $b$-matching problems.

Findings

Data reductions significantly shrink input size.

Initial solutions are competitive on real-world hypergraphs.

Algorithms are practical and effective in experiments.

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.