Search for a moving target
A target starts at unknown distance \(d\) from the origin of a line (or of \(m\) rays) and moves at constant speed \(u\); how fast can robots capture it, depending on what they know about \(d\) and \(u\)?
Problem statement and measuresexpand
Robots of unit speed start at the origin of the line (or of \(m\) rays). The target starts at distance \(d \ge 1\) from the origin, on a side (ray) unknown to the robots, and moves along it at constant speed \(u\), either away from the origin (\(u < 1\), since a faster target can never be caught) or toward it (\(u \ge 0\)); the robots know which. The target cannot change its speed or direction. Depending on the setting the robots know \(d\), \(u\), both or neither. The target is captured when every robot stands on it. Time is the competitive ratio: the supremum, over the target's initial positions and speeds, of the capture time divided by the time the robots would need if they knew the target's initial position.
Measures: Competitive ratio, worst-case capture time divided by the capture time of robots that know the target's initial position.
Line, two face-to-face robots, away, \(u\) known
draftKnowing only the speed, two face-to-face robots capture a target moving away with ratio \((3+u)^2/(1-u)^2\), the single-robot value, using three turns.
Assumed (added on the last edge are outlined)
Not assumed
Formal definitions of this setting
- Domain: line
- The real line; the target moves along one side of the origin.
- Number of robots: two robots
- Two robots, both of which must reach the target.
- Direction of the target: away
- The target moves away from the origin at constant speed \(u < 1\), and the robots know it.
- Knowledge of the target: speed known
- The robots know \(u\) but not \(d\).
- Communication: face-to-face
- Face-to-face: two robots exchange information only when they are at the same point at the same time.
- Faults: no faults
- Every robot detects and reports the target.
- Speeds: equal speeds
- All robots move at most at unit speed.
- Randomness: deterministic
- Trajectories are fixed functions of the information received.
What is known
- bound for robots that know \(d\) and \(u\) (Theorem 2.2 of Jawhar & Kranakis 2026b)
- the argument does not use the communication model
Results, in order of publication
Introductory and best: each robot zigzags on its own side (Algorithm 4, Theorem 3.1, unboundedly many turns), or both walk apart at speed \((3u+1)/(3+u)\) and the finder fetches the other (Algorithm 5, Theorem 3.3, three turns, the fewest possible by Theorem 3.4).
Open questions
Is \((3+u)^2/(1-u)^2\) optimal for two face-to-face robots for every \(u \in (0,1)\), as it is for one robot?
Notes
At \(u = 0\) this is evacuation of two face-to-face robots on the line, where 9 is optimal Chrobak et al. 2015.
What carries over from neighbouring settings
Read off the order of assumptions, not checked case by case: results may carry extra assumptions (shown on their pages), and asynchronous time is measured in epochs.
Algorithms from weaker settings work here
Lower bounds from stronger settings hold here
Know a better result for this setting, or found a mistake? Suggest a result · Report an error. Accepted contributions are acknowledged.