Search and evacuation with a near majority of faulty agents

Abstract

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.