Towards Optimal Degree-Distributions for Left-Perfect Matchings in Random Bipartite Graphs

Abstract

Consider a random bipartite multigraph G with n left nodes and m ≥ n ≥ 2 right nodes. Each left node x has d _x  ≥ 1 random right neighbors. The average left degree $\bar{{\mathrm{\scriptstyle\Delta}}}$ is fixed, $\bar{{\mathrm{\scriptstyle\Delta}}}\geq2$. 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 $\bar{{\mathrm{\scriptstyle\Delta}}}$ is an integer then it is optimal to use a fixed degree of $\bar{{\mathrm{\scriptstyle\Delta}}}$ for all left nodes. If $\bar{{\mathrm{\scriptstyle\Delta}}}$ is non-integral then an optimal degree-distribution has the property that each left node x has two possible degrees, $\ensuremath{\lfloor \bar{{\mathrm{\scriptstyle\Delta}}}\rfloor}$ and $\ensuremath{\lceil \bar{{\mathrm{\scriptstyle\Delta}}}\rceil}$, 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 $\ensuremath{\lceil \bar{{\mathrm{\scriptstyle\Delta}}}\rceil}-\bar{{\mathrm{\scriptstyle\Delta}}}$. Furthermore, if n = c·m and $\bar{{\mathrm{\scriptstyle\Delta}}}>2$ is constant, then each distribution of the left degrees that meets the conditions above determines the same threshold $c^*({\bar{{\mathrm{\scriptstyle\Delta}}}})$ that has the following property as n goes to infinity: If $c<c^*({\bar{{\mathrm{\scriptstyle\Delta}}}})$ then there exists a left-perfect matching with high probability. If $c>c^*({\bar{{\mathrm{\scriptstyle\Delta}}}})$ then there exists no left-perfect matching with high probability. The threshold $c^*({\bar{{\mathrm{\scriptstyle\Delta}}}})$ is the same as the known threshold for offline k-ary cuckoo hashing for integral or non-integral $k=\bar{{\mathrm{\scriptstyle\Delta}}}$.

Research supported by DFG grant DI 412/10-2.