The problem of finding a minimum-weight connected dominating set (CDS) of a given undirected graph has been studied actively, motivated by operations of wireless ad hoc networks. In this paper, we formulate a new stochastic variant of the problem. In this problem, each node in the graph has a hidden random state, which represents whether the node is active or inactive, and we seek a CDS of the graph that consists of the active nodes. We consider an adaptive algorithm for this problem, which repeat choosing nodes and observing the states of the nodes around the chosen nodes until a CDS is found. Our algorithms have a theoretical performance guarantee that the sum of the weights of the nodes chosen by the algorithm is at most $O(\alpha \log (1/\delta))$ times that of any adaptive algorithm in expectation, where $\alpha $ is an approximation factor for the node-weighted polymatroid Steiner tree problem and $\delta $ is the minimum probability of possible scenarios on the node states.

中文翻译：

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在不确定图中寻找连通控制集的自适应算法

在无线自组织网络的操作的推动下，已经积极研究了寻找给定无向图的最小权重连接控制集（CDS）的问题。在本文中，我们制定了该问题的新的随机变体。在此问题中，图中的每个节点都有一个隐藏的随机状态，代表该节点处于活动状态还是非活动状态，我们寻求由活动节点组成的图形的CDS。我们考虑针对该问题的自适应算法，该算法重复选择节点并观察所选节点周围的节点状态，直到找到CDS。我们的算法具有理论上的性能保证，即算法选择的节点的权重之和最大为 $ O（\ alpha \ log（1 / \ delta））$ 乘以预期中任何自适应算法的乘积，其中 $ \ alpha $ 是节点加权多拟Steiner树问题的近似因子，并且 $ \ delta $ 是节点状态上可能的方案的最小概率。