Probabilistically Faulty Searching on a Half-Line

We study $p$-Faulty Search, a variant of the classic cow-path optimization problem, where a unit speed robot searches the half-line (or $1$-ray) for a hidden item. The searcher is probabilistically faulty, and detection of the item with each visitation is an independent Bernoulli trial whose probability of success $p$ is known. The objective is to minimize the worst case expected detection time, relative to the distance of the hidden item to the origin. A variation of the same problem was first proposed by Gal in 1980. Then in 2003, Alpern and Gal [The Theory of Search Games and Rendezvous] proposed a so-called monotone solution for searching the line ($2$-rays); that is, a trajectory in which the newly searched space increases monotonically in each ray and in each iteration. Moreover, they conjectured that an optimal trajectory for the $2$-rays problem must be monotone. We disprove this conjecture when the search domain is the half-line ($1$-ray). We provide a lower bound for all monotone algorithms, which we also match with an upper bound. Our main contribution is the design and analysis of a sequence of refined search strategies, outside the family of monotone algorithms, which we call $t$-sub-monotone algorithms. Such algorithms induce performance that is strictly decreasing with $t$, and for all $p \in (0,1)$. The value of $t$ quantifies, in a certain sense, how much our algorithms deviate from being monotone, demonstrating that monotone algorithms are sub-optimal when searching the half-line.Comment: This is full version of the paper with the same title which will appear in the proceedings of the 14th Latin American Theoretical Informatics Symposium (LATIN20), Sao Paulo, Brazil, May 25-29, 202