Jean-Sébastien Sereni

Collaborative search on the plane without communication

We use distributed computing tools to provide a new perspective on the behavior of cooperative biological ensembles. We introduce the Ants Nearby Treasure Search (ANTS) problem, a generalization of the classical cow-path problem [10, 20, 41, 42], which is relevant for collective foraging in animal groups. In the ANTS problem, 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. We focus 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 the 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 Ω(D + D2/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 a tight bound for the competitive penalty that must be paid, in the running time, if the agents have no information about k. Specifically, this bound is slightly more than logarithmic in the number of agents. In addition, we give a lower bound for the setting in which the agents are given some estimation of 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 use some information regarding k. Finally, we propose a uniform algorithm that is both efficient and extremely simple, suggesting its relevance for actual biological scenarios.

Total-Coloring of Plane Graphs with Maximum Degree Nine

The central problem of the total-colorings is the total-coloring conjecture, which asserts that every graph of maximum degree

A New Lower Bound Based on Gromov’s Method of Selecting Heavily Covered Points

\Delta

Toward more localized local algorithms: removing assumptions concerning global knowledge

admits a

Fullerene graphs have exponentially many perfect matchings

(\Delta+2)

Griggs and Yeh's Conjecture and

-total-coloring. Similar to edge-colorings—with Vizings edge-coloring conjecture—this bound can be decreased by 1 for plane graphs of higher maximum degree. More precisely, it is known that if

L(p,1)

\Delta\ge10

-labelings

, then every plane graph of maximum degree

Identifying and locating–dominating codes in (random) geometric networks

\Delta

A new bound for the 2/3 conjecture

is