We study the Feedback Vertex Set and the Vertex Cover problem in a natural variant of the classical online model that allows for delayed decisions and reservations. Both problems can be characterized by an obstruction set of subgraphs that the online graph needs to avoid. In the case of the Vertex Cover problem, the obstruction set consists of an edge (i.e., the graph of two adjacent vertices), while for the Feedback Vertex Set problem, the obstruction set contains all cycles. In the delayed-decision model, an algorithm needs to maintain a valid partial solution after every request, thus allowing it to postpone decisions until the current partial solution is no longer valid for the current request. The reservation model grants an online algorithm the new and additional option to pay a so-called reservation cost for any given element in order to delay the decision of adding or rejecting it until the end of the instance. For the Feedback Vertex Set problem, we first analyze the variant with only delayed decisions, proving a lower bound of $4$ and an upper bound of $5$ on the competitive ratio. Then we look at the variant with both delayed decisions and reservation. We show that given bounds on the competitive ratio of a problem with delayed decisions impliy lower and upper bounds for the same problem when adding the option of reservations. This observation allows us to give a lower bound of $\min{\{1+3\alpha,4\}}$ and an upper bound of $\min{\{1+5\alpha,5\}}$ for the Feedback Vertex Set problem. Finally, we show that the online Vertex Cover problem, when both delayed decisions and reservations are allowed, is $\min{\{1+2\alpha, 2\}}$-competitive, where $\alpha \in \mathbb{R}_{\geq 0}$ is the reservation cost per reserved vertex.

翻译：我们研究反馈顶点集和顶点覆盖问题在经典在线模型的一种自然变体中的表现，该变体允许延迟决策和保留。这两个问题均可通过在线图需要避免的子图障碍集来刻画。对于顶点覆盖问题，障碍集包含一条边（即两个相邻顶点构成的图），而对于反馈顶点集问题，障碍集包含所有环。在延迟决策模型中，算法需要在每次请求后维持一个有效的部分解，从而允许其将决策推迟到当前部分解对当前请求不再有效时。保留模型赋予在线算法一项新的额外选项：为任意元素支付所谓的保留成本，以便将其添加或拒绝的决策推迟到实例末尾。针对反馈顶点集问题，我们首先分析仅含延迟决策的变体，证明竞争比的下界为4、上界为5。随后研究同时包含延迟决策和保留的变体。我们证明，对于具有延迟决策的问题的竞争比界限，可以推导出同一问题在引入保留选项后的下界与上界。这一观察使我们能够给出反馈顶点集问题的下界$\min{\{1+3\alpha,4\}}$和上界$\min{\{1+5\alpha,5\}}$。最后，我们证明当允许同时使用延迟决策和保留时，在线顶点覆盖问题具有$\min{\{1+2\alpha, 2\}}$-竞争比，其中$\alpha \in \mathbb{R}_{\geq 0}$为每个保留顶点的保留成本。