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. In this paper, we introduce a natural generalization of this game in which $k$ random vertices $u_1, \ldots, u_k$ are presented to the player in each round. She needs to select one of the presented vertices and connect to any vertex she wants. We focus on the following three monotone properties: minimum degree at least $\ell$, the existence of a perfect matching, and the existence of a Hamiltonian cycle.

翻译：半随机图过程是一个单人游戏，其中玩家初始时拥有一个包含$n$个顶点的空图。在每一轮中，一个顶点$u$独立均匀随机地呈现给玩家。玩家随后自适应地选择一个顶点$v$，并将边$uv$添加到图中。对于固定的单调图性质，玩家的目标是在尽可能少的轮数内，以高概率迫使图满足该性质。在本文中，我们引入该游戏的一个自然推广：在每一轮中，有$k$个随机顶点$u_1, \ldots, u_k$呈现给玩家。玩家需要从这些呈现的顶点中选择一个，并将其连接到任意她选择的顶点。我们重点关注以下三个单调性质：最小度至少为$\ell$、存在完美匹配以及存在哈密顿圈。