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

Improper choosability of planar graphs has been widely studied. In particular,Škrekovski investigated the smallest integer g k such that every planar graph of girth at least g k is k-improper 2-choosable. He proved [9] that 6≤ g 1 ≤9; 5≤ g 2 ≤ 7; 5 ≤ g 3 ≤ 6; and ∀k ≥ 4, g k = 5. In this article, we study the greatest real M(k, l) such that every graph of maximum average degree less than. As a corollary, we deduce that g 1 ≤ 8 and g 2 ≤ 6, and we obtain new results for graphs of higher genus. We also provide an upper bound for M (k, l). This implies that for any fixed l, M(k, l) −→ k→∞ 2l.

International audienceNumerous sophisticated local algorithm were suggested in the literature for various fundamental problems. Notable examples are the MIS and (∆+1)-coloring algorithms by Barenboim and Elkin [6], by Kuhn [22], and by Panconesi and Srinivasan [34], as well as the O(∆ 2)-coloring algorithm by Linial [28]. Unfortunately, most known local algorithms (including, in particular, the aforementioned algorithms) are non-uniform, that is, local algorithms generally use good estimations of one or more global parameters of the network, e.g., the maximum degree ∆ or the number of nodes n. This paper provides a method for transforming a non-uniform local algorithm into a uniform one. Furthermore , the resulting algorithm enjoys the same asymp-totic running time as the original non-uniform algorithm. Our method applies to a wide family of both deterministic and randomized algorithms. Specifically, it applies to almost all state of the art non-uniform algorithms for MIS and Maximal Matching, as well as to many results concerning the coloring problem. (In particular, it applies to all aforementioned algorithms.) To obtain our transformations we introduce a new distributed tool called pruning algorithms, which we believe may be of independent interest