Perfect Matchings in the Semi-random Graph Process

The semi-random graph process is a single player game in which the player is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then adaptively selects a vertex $v$, and adds the edge $uv$ to the graph. For a fixed monotone graph property, the objective of the player is to force the graph to satisfy this property with high probability in as few rounds as possible. We focus on the problem of constructing a perfect matching in as few rounds as possible. In particular, we present an adaptive strategy for the player which achieves a perfect matching in $\beta n$ rounds, where the value of $\beta < 1.206$ is derived from a solution to some system of differential equations. This improves upon the previously best known upper bound of $(1+2/e+o(1)) \, n < 1.736 \, n$ rounds. We also improve the previously best lower bound of $(\ln 2 + o(1)) \, n > 0.693 \, n$ and show that the player cannot achieve the desired property in less than $\alpha n$ rounds, where the value of $\alpha > 0.932$ is derived from a solution to another system of differential equations. As a result, the gap between the upper and lower bounds is decreased roughly four times.