The power of pheromones in ant foraging

Consider the following problem. An unknown number k of ants, initially located at the nest, searches the plane for a treasure. The identical, deterministic ants take steps asynchronously, until eventually an ant moves onto the location of the treasure. The only means to record the progress of the search is by marking each explored grid point with a pheromone. It is well-known that ants have this capability, as well as being able to determine the direction in which the nest lies. Under these assumptions, we provide an extremely simple algorithm that is factor 5.5-optimal in terms of the worst-case number of asynchronous rounds required to find the treasure. This shows that the proposed mechanism of marking searched areas by pheromone is very powerful: Feinerman et al. proved that, without it, ants need to be given a constant-factor approximation of k to achieve asymptotic optimality, even when randomization is available.