Collaborative search on the plane without communication

We generalize the classical cow-path problem [7, 14, 38, 39] into a question that is relevant for collective foraging in animal groups. Specifically, we consider a setting in which k identical (probabilistic) agents, initially placed at some central location, collectively search for a treasure in the two-dimensional plane. The treasure is placed at a target location by an adversary and the goal is to find it as fast as possible as a function of both k and D, where D is the distance between the central location and the target. This is biologically motivated by cooperative, central place foraging such as performed by ants around their nest. In this type of search there is a strong preference to locate nearby food sources before those that are further away. Our focus is on trying to find what can be achieved if communication is limited or altogether absent. Indeed, to avoid overlaps agents must be highly dispersed making communication difficult. Furthermore, if agents do not commence the search in synchrony then even initial communication is problematic. This holds, in particular, with respect to the question of whether the agents can communicate and conclude their total number, k. It turns out that the knowledge of k by the individual agents is crucial for performance. Indeed, it is a straightforward observation that the time required for finding the treasure is $\Omega$(D + D 2 /k), and we show in this paper that this bound can be matched if the agents have knowledge of k up to some constant approximation. We present an almost tight bound for the competitive penalty that must be paid, in the running time, if agents have no information about k. Specifically, on the negative side, we show that in such a case, there is no algorithm whose competitiveness is O(log k). On the other hand, we show that for every constant $\epsilon \textgreater{} 0$, there exists a rather simple uniform search algorithm which is $O( \log^{1+\epsilon} k)$-competitive. In addition, we give a lower bound for the setting in which agents are given some estimation of k. As a special case, this lower bound implies that for any constant $\epsilon \textgreater{} 0$, if each agent is given a (one-sided) $k^\epsilon$-approximation to k, then the competitiveness is $\Omega$(log k). Informally, our results imply that the agents can potentially perform well without any knowledge of their total number k, however, to further improve, they must be given a relatively good approximation of k. Finally, we propose a uniform algorithm that is both efficient and extremely simple suggesting its relevance for actual biological scenarios.

💡 Research Summary

The paper extends the classic cow‑path problem to a multi‑agent setting that is motivated by central‑place foraging in social insects. A group of k identical probabilistic agents starts from a common nest and must locate a treasure placed at an adversarial distance D in the Euclidean plane. Communication among agents is either absent or severely limited, so each agent must act based on its own information and possibly a rough estimate of the total number of agents. The performance measure is the competitive ratio: the worst‑case running time divided by the information‑theoretic lower bound Ω(D + D²/k).

First, the authors prove that this lower bound holds regardless of any knowledge about k. The term D reflects the unavoidable travel time to the treasure, while D²/k captures the time needed for the agents collectively to sweep an area of radius D when they cannot coordinate.

When each agent knows a constant‑factor approximation of k (i.e., it knows a value (\hat k) such that (c_1k \le \hat k \le c_2k) for fixed constants (c_1,c_2)), the paper presents a “partitioned spiral” algorithm that matches the lower bound. The algorithm proceeds in exponentially increasing radius phases (r_i = 2^i \cdot (D/\hat k)). In phase i the agents are assigned distinct angular sectors and each follows a spiral path of radius (r_i). Because the sectors are disjoint, overlap is avoided and the total area covered after i phases is Θ(k·r_i). The number of phases needed is O(log(D·\hat k/D)), leading to a total time Θ(D + D²/k). Thus a constant‑factor estimate of k is sufficient to achieve optimal performance.

In contrast, if agents have no information about k (the uniform setting), the authors show a strong negative result: no algorithm can achieve an O(log k) competitive ratio. The proof uses an information‑theoretic argument that any algorithm must distinguish between instances where the true k lies in adjacent exponential intervals, and without any estimate this requires an extra logarithmic factor.

Nevertheless, the paper provides a simple uniform algorithm that attains an O(log^{1+ε} k) competitive ratio for any constant ε > 0. Each agent independently performs a sequence of phases with radii doubling each time, choosing a random direction at the start of each phase. By extending the number of phases by a factor of log^ε k, the algorithm ensures that with high probability the agents collectively cover the necessary area, incurring only a sub‑logarithmic penalty.

The authors also establish a lower bound for the case where agents receive a one‑sided k^ε‑approximation of the true k. Even this relatively coarse estimate forces any algorithm to suffer a Ω(log k) competitive penalty, showing that a constant‑factor approximation is essentially the threshold for optimality.

Biologically, the results suggest that social insects do not need precise knowledge of colony size to forage efficiently; a rough sense of scale (within a constant factor) suffices to coordinate a near‑optimal search, while completely ignorant colonies can still perform reasonably well at the cost of a modest logarithmic slowdown. The proposed uniform algorithm is extremely simple, making it a plausible model for real animal behavior where communication is limited and agents start asynchronously.

Finally, the paper outlines future directions, including handling asynchronous start times more rigorously, extending the model to environments with obstacles or varying terrain, and validating the theoretical predictions with empirical studies of insect foraging. Overall, the work bridges algorithmic theory and biological collective behavior, demonstrating how limited information and lack of communication shape the efficiency of collaborative search on the plane.