# The ANTS problem

## TL;DR

This paper introduces the ANTS problem modeling ant-like cooperative search without communication, analyzing the relationship between initial knowledge, memory, and search efficiency, and establishing bounds on these factors.

## Contribution

It provides the first non-trivial lower bounds on memory complexity for probabilistic search agents and connects initial knowledge with search time efficiency.

## Key findings

- Log log k + Θ(1) bits of initial info suffice for optimal search time.
- Memory capacity of Ω(log k) states is necessary for faster search times.
- First non-trivial lower bounds on memory for probabilistic search agents.

## Abstract

We introduce the Ants Nearby Treasure Search (ANTS) problem, which models natural cooperative foraging behavior such as that performed by ants around their nest. In this problem, k probabilistic agents, initially placed at a central location, collectively search for a treasure on the two-dimensional grid. The treasure is placed at a target location by an adversary and the agents' goal is to find it as fast as possible as a function of both k and D, where D is the (unknown) distance between the central location and the target. We concentrate on the case in which agents cannot communicate while searching. It is straightforward to see that the time until at least one agent finds the target is at least $\Omega$(D + D 2 /k), even for very sophisticated agents, with unrestricted memory. Our algorithmic analysis aims at establishing connections between the time complexity and the initial knowledge held by agents (e.g., regarding their total number k), as they commence the search. We provide a range of both upper and lower bounds for the initial knowledge required for obtaining fast running time. For example, we prove that log log k + $\Theta$(1) bits of initial information are both necessary and sufficient to obtain asymptotically optimal running time, i.e., O(D +D 2 /k). We also we prove that for every 0 \textless{} \textless{} 1, running in time O(log 1-- k $\times$(D +D 2 /k)) requires that agents have the capacity for storing $\Omega$(log k) different states as they leave the nest to start the search. To the best of our knowledge, the lower bounds presented in this paper provide the first non-trivial lower bounds on the memory complexity of probabilistic agents in the context of search problems. We view this paper as a "proof of concept" for a new type of interdisciplinary methodology. To fully demonstrate this methodology, the theoretical tradeoff presented here (or a similar one) should be combined with measurements of the time performance of searching ants.