There are $n\geq 3$ unit speed mobile agents placed at the origin of the infinite line. In as little time as possible, the agents must find and evacuate from an exit placed at an initially unknown location on the line. The agents can communicate in the wireless mode in order to facilitate the evacuation (i.e. by announcing the target's location when it is found). However, among the agents are a subset of at most $f$ crash faulty agents who may fail to announce the target when they visit its location. In this paper we study this aforementioned problem for the specific case that $n=2f+1$. We introduce a novel type of search algorithm and analyze its competitive ratio -- the supremum, over all possible target locations, of the ratio of the time the agents take to evacuate divided by the initial distance between the agents and the target. In particular, we demonstrate that the competitive ratio of evacuation is at most $7.437011$ for $(n,f)=(3,1)$; at most $7.253767$ for $(n,f)=(5,2)$ and $(7,3)$; and at most $7.147026$ for $(n,f)=(9,4)$. For larger values of $n=2f+1$ we prove an asymptotic upper bound of $4+2\sqrt{2}$. We also adapt our evacuation algorithm for $(n,f)=(3,1)$ to the problem of search by three agents with one byzantine fault, i.e. the faulty agent may also lie about finding the target. In doing so we improve the best known upper bound on this search problem from 8.653055 to 7.437011.

翻译：设有 $n\geq 3$ 个单位速度移动代理体位于无限直线的原点。代理体需在尽可能短的时间内，找到并疏散至位于直线上初始位置未知的出口。代理体之间可通过无线模式通信以协助疏散（例如在发现目标位置时进行宣告）。然而，代理体中存在至多 $f$ 个崩溃故障代理体，这些故障代理体在访问目标位置时可能无法宣告目标。本文针对 $n=2f+1$ 这一特殊情形研究上述问题。我们引入一种新型搜索算法，并分析其竞争比——即所有可能目标位置下，代理体完成疏散所需时间与代理体初始距离目标之间比值的上确界。具体而言，我们证明：当 $(n,f)=(3,1)$ 时疏散竞争比至多为 $7.437011$；当 $(n,f)=(5,2)$ 和 $(7,3)$ 时至多为 $7.253767$；当 $(n,f)=(9,4)$ 时至多为 $7.147026$。对于更大的 $n=2f+1$ 值，我们证明渐近上界为 $4+2\sqrt{2}$。我们还将 $(n,f)=(3,1)$ 的疏散算法推广至存在一个拜占庭故障代理体的三代理搜索问题（即故障代理体可能对是否找到目标说谎）。通过这一改进，我们将该搜索问题的最佳已知上界从 $8.653055$ 降低至 $7.437011$。