Sharp Thresholds in Adaptive Random Graph Processes

Suppose that $K_n$ is the complete graph on vertex set $[n]$, and $\D$ is a distribution on subsets of its edges. The $\mathcal{D}$-adaptive random graph process (or $\mathcal{D}$-process) is a single player game in which the player is initially presented the empty graph on $[n]$. In each step, a subset of edges of $K_n$, say $X$, is independently sampled from $\mathcal{D}$ and presented to the player. The player then adaptively selects precisely one edge $e$ from $X$, and adds $e$ to its current graph. For a fixed (edge) monotone graph property, the objective of the player is to force the graph to satisfy the property in as few steps as possible. Through appropriate choices of $\mathcal{D}$, the $\mathcal{D}$-process generalizes well-studied adaptive processes, such as the Achlioptas process and the semi-random graph process. We prove a theorem which gives a sufficient condition for the existence of a sharp threshold for the property $\mathcal{P}$ in the $\mathcal{D}$-process. We apply our theorem to the semi-random graph process and prove the existence of a sharp threshold when $\mathcal{P}$ corresponds to being Hamiltonian or to containing a perfect matching. These are the first results for the semi-random graph process which show the existence of a sharp threshold when $\mathcal{P}$ corresponds to containing a \textit{sparse} spanning graph. Using a separate analytic argument, we show that each sharp threshold is of the form $C_{\mathcal{P}}n$ for some fixed constant $C_{\mathcal{P}}>0$. This answers two of the open problems proposed by Ben-Eliezer et al. (SODA 2020) in the affirmative. Unlike similar results which establish sharp thresholds for certain distributions and properties, we establish the existence of sharp thresholds without explicitly identifying asymptotically optimal strategies.