In the minimum latency problem, an undirected connected graph and a root node together with non-negative edge distances are given to an agent. The agent looks for a tour starting at the root node and visiting all the nodes to minimise the sum of the latencies of the nodes, where the latency of a node is the distance from the root node to the node at its first visit on the tour by the agent. We study an online variant of the problem, where there are $k$ blocked edges in the graph which are not known to the agent in advance. A blocked edge is learned online when the agent arrives at one of its end-nodes. Furthermore, we investigate another online variant of the minimum latency problem involving $k$ blocked edges where each node is associated with a weight to express its priority and the objective is to minimise the summation of the weighted latency of the nodes. In this paper, we prove that the lower bound of $2k+1$ on the competitive ratio of deterministic online algorithms is tight for both weighted and non-weighted variations by introducing an optimal deterministic online algorithm which meets this lower bound. We also present a lower bound of $k+1$ on the expected competitive ratio of randomized online algorithms for both problems. We then develop two polynomial time heuristic algorithms to solve these online problems. We test our algorithms on real life as well as randomly generated instances that are partially adopted from the literature.

在最小延迟问题中，将无向连通图和根节点以及非负边距离一起提供给代理。代理寻找从根节点开始并访问所有节点的旅程，以最小化节点的延迟总和，其中节点的延迟是在旅程中第一次访问时从根节点到节点的距离由代理。我们研究了该问题的在线变体，其中有$克$图中被阻止的边，这些边是代理事先不知道的。当代理到达其终端节点之一时，在线学习被阻塞的边缘。此外，我们研究了最小延迟问题的另一个在线变体，涉及$克$阻塞边，其中每个节点都与一个权重相关联以表示其优先级，目标是最小化节点加权延迟的总和。在本文中，我们证明了下界$2克+1$通过引入满足此下界的最佳确定性在线算法，确定性在线算法的竞争比率对于加权和非加权变化都是严格的。我们还提出了一个下界$克+1$关于这两个问题的随机在线算法的预期竞争率。然后我们开发了两个多项式时间启发式算法来解决这些在线问题。我们在现实生活中以及从文献中部分采用的随机生成的实例中测试我们的算法。