Linear Search for an Escaping Target with Unknown Speed

TL;DR

This work advances the theory of linear search for an escaping target with unknown speed by quantifying how knowledge about the target's starting distance $d$ and speed $v$ affects the competitive ratio. It provides a tight lower bound showing no strategy can achieve $CR(u)=O(u^{4-\varepsilon})$ when $d$ is known, and develops an upper-bound algorithm $\mathcal{A}_1$ achieving $CR(u) \le 56.18\,u^{4-(\log_2\log_2 u)^{-2}}$ (with $CR=9$ for small $u$), effectively optimal up to lower-order terms. For the no-knowledge scenario, it introduces a strategy $\mathcal{A}_2$ yielding improved bounds $CR_{\mathcal{A}_2}(u,d) \le 1+\frac{1}{d}\big(56.18(ud)^{4-(\log_2\log_2 (ud))^{-2}}-1\big)$ for $ud>4$, and $CR \le 1+8/d$ for $ud\le4$, extending to cases where $d$ is unknown. The paper also shows a reduction to the scaled case $CR_{\mathcal{A}}(u,d) \le CR_{\mathcal{A}}(ud,1)$ when both $u$ and $d$ are unknown, enabling stronger bounds and addressing an open problem from prior work. Overall, the results push the boundary on knowledge-dependent competitive ratios in linear search and open avenues for multi-agent extensions with communication constraints.

Abstract

We consider linear search for an escaping target whose speed and initial position are unknown to the searcher. A searcher (an autonomous mobile agent) is initially placed at the origin of the real line and can move with maximum speed $1$ in either direction along the line. An oblivious mobile target that is moving away from the origin with an unknown constant speed $v<1$ is initially placed by an adversary on the infinite line at distance $d$ from the origin in an unknown direction. We consider two cases, depending on whether $d$ is known or unknown. The main contribution of this paper is to prove a new lower bound and give algorithms leading to new upper bounds for search in these settings. This results in an optimal (up to lower order terms in the exponent) competitive ratio in the case where $d$ is known and improved upper and lower bounds for the case where $d$ is unknown. Our results solve an open problem proposed in [Coleman et al., Proc. OPODIS 2022].

Theorems & Definitions (31)

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