Collaborative search on the plane without communication

Abstract

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

A Cooperative Switching Algorithm for Multi-Agent Foraging

“…Ants Nearby Treasure Search (ANTS) is a problem in which k identical agents, initially placed at some central location, collectively search for a treasure in a two-dimensional plane. This problem can be resolved by a collection of identical ants that does not communicate between each other [31]. Some works present a survey on online algorithms for searching and exploration in the plan [32].…”

Linear Search by a Pair of Distinct-Speed Robots

“…The performance of an online algorithm is measured by its competitive ratio, i.e., the worst-case ratio of its cost with respect to the offline cost, which is the search time of the optimal algorithm with full a priori knowledge of the environment and the target placement. Many search problems, especially for geometric environments, are analyzed from this perspective, in particular when the cost of the offline solution is just the distance to the target; see [4,13,25,30].…”

“…However, optimization of the search by the use of multiple robots often involves coordination issues, where the searchers need to communicate in order to synchronize their efforts and adequately split the entire task into portions assigned to individual robots (cf. [13,23,25,27]). As this objective is often not easy to achieve, some multi-robot search problems turn out to be NP-hard (e.g., see [27]).…”

Stigmergic MASA: A Stigmergy Based Algorithm for Multi-Target Search

“…A direct communication via WIFI model is used between robots and their neighbors. The paper [11], introduce the ANTS (Ants Nearby Treasure Search) problem, in which k identical agents, initially placed at some central location, collectively search for a treasure in a two-dimensional plane, without any communication between them. A survey of online algorithms for searching and exploration in the plane is given in [34].…”

“…The random walk is the best option when there is some degree of uncertainty in the environment and a reduced perceptual capabilities [10] because it is simple, needs no memory and self-stabilizes. However, it is inefficient in a two-dimensional infinite grid, where it results in an infinite searching time, even if the target is nearby [11], it results also in energy consumption and malfunction risks. To deal with these limits, some effective ways to coordinate multiple agents in their searching task need to take place.…”