无置换半随机过程
Semi-random process without replacement
摘要：
半随机过程涉及一个自适应决策者，其目标是在在线随机环境中实现某些预定目标。我们引入并研究了一个半随机多重图过程，它形成了Ben-Eliezer，Hefetz，Kronenberg，Parczyk，Shikhelman和Stojaković（2020）引入的过程的无替换变体。该过程从顶点集$[n]$上的空图开始。对于每一个正整数$q$和$1\leq r\leq n$，在该过程的第$((q-1)n+r)$轮中，称为\emph{Builder}的决策者被提供顶点$π_q(r)$，其中$π_1, π_2, \ldots$是$S_n$中的排列序列，其被独立地且均匀地随机选择。然后，Builder选择一个额外的顶点（根据他选择的策略），并通过一条边将其连接到$π_q(r)$。对于几个自然图的属性，如$k$ -连通性，最小度至少$k$，并建立一个给定的生成图（标记或未标记），我们确定典型的轮数生成器需要，以便构建一个具有所需属性的图。一路上，我们介绍和分析一个骨灰盒模型，也可能有独立的利益。
英文摘要：
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 Stojaković (2020). The process starts with an empty graph on the vertex set $[n]$. For every positive integers $q$ and $1\leq r\leq n$, in the $((q-1)n+r)$th round of the process, the decision-maker, called \emph{Builder}, is offered the vertex $π_q(r)$, where $π_1, π_2, \ldots$ is a sequence of permutations in $S_n$, 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 $π_q(r)$. 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.
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