The $\mathcal{D}$-process is a single player game in which the player is initially presented the empty graph on $n$ vertices. In each step, a subset of edges $X$ is independently sampled according to a distribution $\mathcal{D}$. The player then selects one edge $e$ from $X$, and adds $e$ to its current graph. For a fixed monotone increasing graph property $\mathcal{P}$, the objective of the player is to force the graph to satisfy $\mathcal{P}$ in as few steps as possible. Through appropriate choices of $\mathcal{D}$, the $\mathcal{D}$-process generalizes well-studied adaptive random graph processes, such as the Achlioptas process and the semi-random graph process We prove a sufficient condition for the existence of a sharp threshold for $\mathcal{P}$ in the $\mathcal{D}$-process. For the semi-random process, we use this condition to 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 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.

翻译：$\mathcal{D}$-过程是一种单人游戏，其中玩家初始时得到包含$n$个顶点的空图。在每一步中，边的子集$X$根据分布$\mathcal{D}$独立采样。然后玩家从$X$中选择一条边$e$，并将其添加到当前图中。对于固定的单调递增图性质$\mathcal{P}$，玩家的目标是尽可能少的步数迫使图满足$\mathcal{P}$。通过适当选择$\mathcal{D}$，$\mathcal{D}$-过程概括了研究较充分的适应性随机图过程，如Achlioptas过程和半随机图过程。我们证明了$\mathcal{D}$-过程中$\mathcal{P}$存在尖锐阈值的充分条件。对于半随机过程，我们利用该条件证明，当$\mathcal{P}$对应于图具有哈密顿性或包含完美匹配时，存在尖锐阈值。这是首个针对半随机图过程的结果，表明当$\mathcal{P}$对应于包含稀疏生成图时存在尖锐阈值。通过单独的分析论证，我们证明每个尖锐阈值形如$C_{\mathcal{P}}n$，其中$C_{\mathcal{P}}>0$为固定常数。这肯定了Ben-Eliezer等人(SODA 2020)提出的两个开放问题。与某些分布和性质建立尖锐阈值的类似结果不同，我们在未明确确定渐近最优策略的情况下建立了尖锐阈值的存在性。