Linear Search with Probabilistic Detection and Variable Speeds

TL;DR

This paper investigates new variants of the linear search problem where an agent moves at variable speeds and detection probabilities, providing algorithms and bounds for different scenarios and establishing optimality in some cases.

Contribution

It introduces novel variants of the linear search problem with probabilistic detection and variable speeds, along with algorithms and bounds for these cases.

Findings

Algorithms with competitive ratio bounds for different speed and detection probability scenarios.

Proof of optimality for the case with zero detection probability.

Upper bounds on competitive ratios for the new search variants.

Abstract

We present results on new variants of the famous linear search (or cow-path) problem that involves an agent searching for a target with unknown position on the infinite line. We consider the variant where the agent can move either at speed $1$ or at a slower speed $v \in [0, 1)$. When traveling at the slower speed $v$, the agent is guaranteed to detect the target upon passing through its location. When traveling at speed $1$, however, the agent, upon passing through the target's location, detects it with probability $p \in [0, 1]$. We present algorithms and provide upper bounds for the competitive ratios for three cases separately: when $p=0$, $v=0$, and when $p,v \in (0,1)$. We also prove that the provided algorithm for the $p=0$ case is optimal.