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Critical Thresholds for Maximum Cardinality Matching on General Hypergraphs

Christopher Sumnicht, Jamison W. Weber, Dhanush R. Giriyan, Arunabha Sen

TL;DR

The Erd\"{o}s-R\'{e}nyi model is extended to general hypergraphs on $n$ vertices and $M$ hyperedges to consider the problem of determining critical thresholds for the largest cardinality matching, and it is shown that for $M=o(1.155^n)$ the size of the maximum cardinality matching is almost surely 1.

Abstract

Significant work has been done on computing the ``average'' optimal solution value for various $\mathsf{NP}$-complete problems using the Erdös-Rényi 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 Erdös-Rényi 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 $M=Θ(2^n)$ then the size of the maximum cardinality matching is $Ω(n^{\frac12-γ})$ for an arbitrary $γ>0$. Lastly, we address the gap where $Ω(1.155^n)=M=o(2^n)$ empirically through computer simulations.

Critical Thresholds for Maximum Cardinality Matching on General Hypergraphs

TL;DR

The Erd\"{o}s-R\'{e}nyi model is extended to general hypergraphs on vertices and hyperedges to consider the problem of determining critical thresholds for the largest cardinality matching, and it is shown that for the size of the maximum cardinality matching is almost surely 1.

Abstract

Significant work has been done on computing the ``average'' optimal solution value for various -complete problems using the Erdös-Rényi 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 Erdös-Rényi model to general hypergraphs on vertices and hyperedges. We consider the problem of determining critical thresholds for the largest cardinality matching, and we show that for the size of the maximum cardinality matching is almost surely 1. On the other hand, if then the size of the maximum cardinality matching is for an arbitrary . Lastly, we address the gap where empirically through computer simulations.
Paper Structure (9 sections, 8 theorems, 41 equations, 2 figures)

This paper contains 9 sections, 8 theorems, 41 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Analysis
  5. Lower Bounds
  6. Empirical Studies
  7. Validation of Convergence to Unity for Small $M$
  8. Empirical Study of the Behavioral Gap
  9. Discussion

Key Result

Theorem 1

Let $(\mathcal{U},\mathcal{S})\sim\mathcal{H}(n,M)$. If $M=o(1.155^n)$, then the hyper-matching number of $(\mathcal{U},\mathcal{S})$ is almost surely 1. If $M=\Theta(2^n)$, then the hyper-matching number of $(\mathcal{U},\mathcal{S})$ is almost surely in $[\Omega(n^{\frac{1}{2}-\delta}),n]$ for arb

Figures (2)

  • Figure 1: A plot of the average (taken over 30 separate trials per value of $n$) and 95% confidence intervals for the exact hyper-matching number given that the number of hyperedges is exactly $\lfloor 1.154^n\rfloor$. The plot shows that the average hyper-matching number remains close to 1 for $n\le 55$.
  • Figure 2: (a) A plot of the average (taken over 30 separate trials per value of $M$) and 95% confidence intervals for the exact hyper-matching number given $n=13$ and $M$ ranging between $1.155^{13}$ and $2^{13}-1$ by increments of $10$. (b) A close up plot of the average (taken over 30 separate trials per value of $M$) and 95% confidence intervals for the exact hyper-matching number given $n=13$ and $M$ ranging between $1$ and $20$. Note that $M=1.155^{13}\approx6$ marks approximately the lower bound of the behavioral gap (shown in red) as predicted by our analysis. Empirically, we observe the abrupt increase at $M=4$ (shown by the dashed red line).

Theorems & Definitions (17)

  • Definition 1
  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 7 more