Semi-random process without replacement

Semi-random processes involve an adaptive decision-maker, whose goal is to achieve some pre-determined objective in an online randomized environment. We introduce and study a semi-random multigraph process, which forms a no-replacement variant of the process that was introduced by Ben-Eliezer, Hefetz, Kronenberg, Parczyk, Shikhelman and Stojakovi\'c (2020). The process starts with an empty graph on the vertex set $[n]$. For every positive integers q and 1≤ r≤ n, in the $((q-1)n+r)$th round of the process, the decision-maker, called Builder, is offered the vertex πq(r), where π₁, π₂, … is a sequence of permutations in Sₙ, chosen independently and uniformly at random. Builder then chooses an additional vertex (according to a strategy of his choice) and connects it by an edge to πq(r). For several natural graph properties, such as k-connectivity, minimum degree at least k, and building a given spanning graph (labeled or unlabeled), we determine the typical number of rounds Builder needs in order to construct a graph having the desired property. Along the way we introduce and analyze an urn model which may also have independent interest.