Optimal Shoreline Search for One and Two Agents

Determine the globally optimal search trajectories and exact worst-case competitive ratios for the classic shoreline search problem in the plane (a half-space boundary search with a fixed starting point), for the cases of (i) a single mobile agent and (ii) two mobile agents. The objective is to identify the optimal strategies and certify their optimality relative to the best known upper and lower bounds reported for these settings.

Background

The shoreline problem is a foundational search problem where a mobile agent starts at a fixed point and seeks an unknown straight boundary (the shoreline) in the plane. Prior work established that for one agent the logarithmic spiral is the best known strategy with competitive ratio 13.81, and for two agents a double logarithmic spiral achieves ratio 5.2644, alongside several lower bounds.

Despite these advances, the exact optimal strategies and competitive ratios for both the one-agent and two-agent shoreline search problems have not been determined. This paper resolves a different variant—the average-case disk-inspection with known distance—but explicitly notes that the optimal solutions for the classic shoreline search with one and two agents remain unresolved.