Critical Thresholds for Maximum Cardinality Matching on General Hypergraphs

TL;DR

This paper investigates the critical thresholds for maximum cardinality matchings in random hypergraphs, extending Erdős-Rényi models, and identifies bounds where the matching size sharply transitions.

Contribution

It extends the Erdős-Rényi model to general hypergraphs and establishes new thresholds for the maximum matching size, including empirical analysis of intermediate regimes.

Findings

For M=o(1.155^n), maximum matching size is almost surely 1.

For M=Θ(2^n), maximum matching size is Ω(n^{0.5-γ}).

Empirical simulations explore the gap between these thresholds.

Abstract

Significant work has been done on computing the ``average'' optimal solution value for various $\mathsf{NP}$-complete problems using the Erd\"{o}s-R\'{e}nyi model to establish \emph{critical thresholds}. Critical thresholds define narrow bounds for the optimal solution of a problem instance such that the probability that the solution value lies outside these bounds vanishes as the instance size approaches infinity. In this paper, we extend the Erd\"{o}s-R\'{e}nyi model to general hypergraphs on $n$ vertices and $M$ hyperedges. We consider the problem of determining critical thresholds for the largest cardinality matching, and we show that for $M=o(1.155^n)$ the size of the maximum cardinality matching is almost surely 1. On the other hand, if $M=\Theta(2^n)$ then the size of the maximum cardinality matching is $\Omega(n^{\frac12-\gamma})$ for an arbitrary $\gamma >0$. Lastly, we address the gap where $\Omega(1.155^n)=M=o(2^n)$ empirically through computer simulations.