树、单圈图与仙人掌图上的在线图探索
Online Graph Exploration on Trees, Unicyclic Graphs and Cactus Graphs
AI总结：
本文研究未知加权图的在线探索问题，证明最近邻算法在树上的竞争比为Θ(log n)，并证明Blocking算法在单圈图和仙人掌图上分别达到3和约3.91的竞争比。
摘要：
我们研究在搜索者初始未知的无向加权图中探索所有顶点的问题。图的边只有在搜索者访问其某个端点时才会被揭示。从某个起始节点出发，搜索者的目标是在返回起始节点之前访问图的每个顶点，并尽可能缩短巡游路径。我们证明，在具有 $n$ 个顶点的树上，最近邻算法的竞争比为 $Θ(\log n)$，即并不优于一般图上的表现。此外，我们考察了此前未被考虑的参数范围内 Blocking 算法的性能，证明它在单圈图上是 3-竞争的，在仙人掌图上是 $5/2+\sqrt{2}\approx 3.91$-竞争的。这两类图目前已知的最优下界为 2。
英文摘要：
We study the problem of exploring all vertices of an undirected weighted graph that is initially unknown to the searcher. An edge of the graph is only revealed when the searcher visits one of its endpoints. Beginning at some start node, the searcher's goal is to visit every vertex of the graph before returning to the start node on a tour as short as possible. We prove that the Nearest Neighbor algorithm's competitive ratio on trees with $n$ vertices is $Θ(\log n)$, i.e. no better than on general graphs. Furthermore, we examine the algorithm Blocking for a range of parameters not considered previously and prove it is 3-competitive on unicyclic graphs as well as $5/2+\sqrt{2}\approx 3.91$-competitive on cactus graphs. The best known lower bound for these two graph classes is 2.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
