Online search with a hint

We introduce the study of search problems, in a setting in which the searcher has some information, or hint concerning the hiding target. In particular, we focus on one of the fundamental problems in search theory, namely the linear search problem. Here, an immobile target is hidden at some unknown position on an unbounded line, and a mobile searcher, initially positioned at some specific point of the line called the root, must traverse the line so as to locate the target. The objective is to minimize the worst-case ratio of the distance traversed by the searcher to the distance of the target from the root, which is known as the competitive ratio of the search.

We consider three settings in regards to the nature of the hint: i) the hint suggests the exact position of the target on the line; ii) the hint suggests the direction of the optimal search (i.e., to the left or the right of the root); and iii) the hint is a general k-bit string that encodes some information concerning the target. Our objective is to study the Pareto-efficiency of strategies in this model, with respect to the tradeoff between consistency and robustness. Namely, we seek optimal, or near-optimal tradeoffs between the searcher's performance if the hint is correct (i.e., provided by a trusted source) and if the hint is incorrect (i.e., provided by an adversary).

We prove several results in each of these three settings. For positional hints, we show that the optimal consistency of r-robust strategies is $(b_r+1)/(b_r-1)$, where $b_r$ is defined to be equal to $\frac{{\rho }_r+\sqrt{{\rho }_r^2-4{\rho }_r}}{2}$, and ${\rho }_r=(r-1)/2$, for all $r\ge 9$. For directional hints, we show that for every $b\ge 1$ and $\delta \in (0,1]$, there exists a strategy with consistency equal to $c=1+2(\frac{b^2}{b^2-1}+\delta \frac{b^3}{b^2-1})$ and robustness equal to $1+2(\frac{b^2}{b^2-1}+\frac{1}{\delta }\frac{b^3}{b^2-1})$; furthermore, we show again that this upper bound is tight. Last, for general k-bit hints, we show upper bounds for general k-bit hints, as well as lower bounds: specifically, we show that the consistency of any 9-robust strategy must be at least 5, and that the consistency of r-robust strategies is at least $1+2b_r/(b_r-1)$, in the case of a natural class of asymptotic strategies.