# A local algorithm and its percolation analysis of bipartite $z$-matching problem

## TL;DR

This paper introduces a local algorithm for the $z$-matching problem on bipartite graphs and develops a mean-field theory to analyze its percolation properties and estimate maximum matching sizes.

## Contribution

It presents a novel local algorithm based on greedy leaf removal and a mean-field percolation theory that improves prediction accuracy over belief propagation for $z$-matching.

## Key findings

- The local algorithm effectively finds large $z$-matchings.
- The mean-field theory accurately estimates $z$-matching sizes on random graphs.
- The framework extends core percolation analysis to hypergraphs with arbitrary degree distributions.

## Abstract

A $z$-matching on a bipartite graph is a set of edges, among which each vertex of two types of the graph is adjacent to at most $1$ and at most $z$ ($\geqslant 1$) edges, respectively. The $z$-matching problem concerns finding $z$-matchings with the maximum size. Our approach to this combinatorial optimization problem is twofold. From an algorithmic perspective, we adopt a local algorithm as a linear approximate solver to find $z$-matchings on any graph instance, whose basic component is a generalized greedy leaf removal procedure in graph theory. From a theoretical perspective, on uncorrelated random bipartite graphs, we develop a mean-field theory for percolation phenomenon underlying the local algorithm, leading to an analytical estimation of $z$-matching sizes on random graphs. Our analytical theory corrects the prediction by belief propagation algorithm at zero-temperature limit in (Krea\v{c}i\'{c} and Bianconi 2019 \textsl{EPL} \textbf{126} 028001). Besides, our theoretical framework extends a core percolation analysis of $k$-XORSAT problems to a general context of uncorrelated random hypergraphs with arbitrary degree distributions of factor and variable nodes.