Reconfiguration schedules, i.e., sequences that gradually transform one solution of a problem to another while always maintaining feasibility, have been extensively studied. Most research has dealt with the decision problem of whether a reconfiguration schedule exists, and the complexity of finding one. A prime example is the reconfiguration of vertex covers. We initiate the study of batched vertex cover reconfiguration, which allows to reconfigure multiple vertices concurrently while requiring that any adversarial reconfiguration order within a batch maintains feasibility. The latter provides robustness, e.g., if the simultaneous reconfiguration of a batch cannot be guaranteed. The quality of a schedule is measured by the number of batches until all nodes are reconfigured, and its cost, i.e., the maximum size of an intermediate vertex cover. To set a baseline for batch reconfiguration, we show that for graphs belonging to one of the classes $\{\mathsf{cycles, trees, forests, chordal, cactus, even\text{-}hole\text{-}free, claw\text{-}free}\}$, there are schedules that use $O(\varepsilon^{-1})$ batches and incur only a $1+\varepsilon$ multiplicative increase in cost over the best sequential schedules. Our main contribution is to compute such batch schedules in $O(\varepsilon^{-1}\log^* n)$ distributed time, which we also show to be tight. Further, we show that once we step out of these graph classes we face a very different situation. There are graph classes on which no efficient distributed algorithm can obtain the best (or almost best) existing schedule. Moreover, there are classes of bounded degree graphs which do not admit any reconfiguration schedules without incurring a large multiplicative increase in the cost at all.

中文翻译：

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分布式顶点覆盖重新配置

重新配置计划，即在始终保持可行性的同时逐渐将问题的一种解决方案转换为另一种解决方案的序列，已被广泛研究。大多数研究都处理了是否存在重新配置时间表的决策问题，以及找到一个的复杂性。一个主要的例子是顶点覆盖的重新配置。我们启动了批量顶点覆盖重新配置的研究，它允许同时重新配置多个顶点，同时要求批处理中的任何对抗性重新配置顺序保持可行性。后者提供鲁棒性，例如，如果不能保证批次的同时重新配置。调度的质量是通过重新配置所有节点之前的批次数及其成本（即中间顶点覆盖的最大大小）来衡量的。为了为批量重新配置设置基线，我们展示了属于以下类别之一的图\text{-}free}\}$，有些调度使用 $O(\varepsilon^{-1})$ 批次，与最佳顺序调度相比，成本仅增加了 $1+\varepsilon$ 乘法增加。我们的主要贡献是在 $O(\varepsilon^{-1}\log^* n)$ 分布式时间中计算这样的批处理计划，我们也证明这是紧张的。此外，我们表明，一旦我们走出这些图类，我们就会面临非常不同的情况。在某些图类上，没有有效的分布式算法可以获得最佳（或几乎最佳）的现有调度。而且，