Towards Optimal Degree-distributions for Left-perfect Matchings in Random Bipartite Graphs

TL;DR

This paper investigates how to choose degree distributions for left nodes in random bipartite graphs to maximize the probability of having a left-perfect matching, revealing optimal strategies depending on whether the average degree is integer or not.

Contribution

It establishes the optimal degree distribution strategies for left nodes in bipartite graphs to maximize matchings, depending on the integrality of the average degree, and links the threshold to cuckoo hashing.

Findings

Fixed degree is optimal when average degree is integer.

For non-integer average degree, a mixed degree distribution is optimal.

The threshold for perfect matchings matches that of cuckoo hashing as graph size grows.

Abstract

Consider a random bipartite multigraph $G$ with $n$ left nodes and $m \geq n \geq 2$ right nodes. Each left node $x$ has $d_x \geq 1$ random right neighbors. The average left degree $\Delta$ is fixed, $\Delta \geq 2$. We ask whether for the probability that $G$ has a left-perfect matching it is advantageous not to fix $d_x$ for each left node $x$ but rather choose it at random according to some (cleverly chosen) distribution. We show the following, provided that the degrees of the left nodes are independent: If $\Delta$ is an integer then it is optimal to use a fixed degree of $\Delta$ for all left nodes. If $\Delta$ is non-integral then an optimal degree-distribution has the property that each left node $x$ has two possible degrees, $\floor{\Delta}$ and $\ceil{\Delta}$, with probability $p_x$ and $1-p_x$, respectively, where $p_x$ is from the closed interval $[0,1]$ and the average over all $p_x$ equals $\ceil{\Delta}-\Delta$. Furthermore, if $n=c\cdot m$ and $\Delta>2$ is constant, then each distribution of the left degrees that meets the conditions above determines the same threshold $c^*(\Delta)$ that has the following property as $n$ goes to infinity: If $c<c^*(\Delta)$ then there exists a left-perfect matching with high probability. If $c>c^*(\Delta)$ then there exists no left-perfect matching with high probability. The threshold $c^*(\Delta)$ is the same as the known threshold for offline $k$-ary cuckoo hashing for integral or non-integral $k=\Delta$.