On Optimal Coverage of a Tree with Multiple Robots

We study the algorithmic problem of optimally covering a tree with $k$ mobile robots. The tree is known to all robots, and our goal is to assign a walk to each robot in such a way that the union of these walks covers the whole tree. We assume that the edges have the same length, and that traveling along an edge takes a unit of time. Two objective functions are considered: the cover time and the cover length. The cover time is the maximum time a robot needs to finish its assigned walk and the cover length is the sum of the lengths of all the walks. We also consider a variant in which the robots must rendezvous periodically at the same vertex in at most a certain number of moves. We show that the problem is different for the two cost functions. For the cover time minimization problem, we prove that the problem is NP-hard when $k$ is part of the input, regardless of whether periodic rendezvous are required or not. For the cover length minimization problem, we show that it can be solved in polynomial time when periodic rendezvous are not required, and it is NP-hard otherwise

Descripción del perfil ontogénico de hormonas esteroides sexuales e intermediarios en testículos del ratón de orejas negras (Peromyscus melanotis, Allen y Chapman, 1897)

Se analizó por radioinmunoanálisis (RIA) el contenido de esteroides sexuales (ES), dos hormonas (progesterona –P4-y testosterona -T-) y tres intermediarios (pregnenolona -P5-; 17a-hidroxi-progesterona –17P4-y androstendiona –A-), en cuatro categorías de edad de Peromyscus melanotis, así como la histología de sus testículos y epidídimos. Cuando se compararon las concentraciones de los ES entre su correspondiente categoría de edad (CE) se encontraron diferencias significativas (P < 0.0001), pero cuando la comparación se hizo entre P5, P4, 17P4 y A en las cuatro CE no hubo diferencias (P > 0.05). Al comparar el contenido de T entre subadultos (CE II) y adultos jóvenes (CE III) la diferencia fue significativa (t1,7 = 30.80, P 0.05); en cambio sí existieron (t1,7 = 14.05, P < 0.0001) cuando la comparación se hizo entre adultos (CE IV) y viejos (CE V). El ciclo ontogénico de la testosterona, así como la espermatogénesis aumentan a partir de la pubertad, alcanzando su máximo en la edad adulta y decrecen progresivamente con el envejecimiento

Separability, Boxicity, and Partial Orders

This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/A collection S = {Si,..., Sn} of disjoint closed convex sets in Rd is separable if there exists a direction (a non-zero vector) −→v of Rd such that the elements of S can be removed, one at a time, by translating them an arbitrarily large distance in the direction −→v without hitting another element of S. We say that Si ≺ Sj if Sj has to be removed before we can remove Si . The relation ≺ defines a partial order P(S, ≺) on S which we call the separability order of S and −→v . A partial order P(X, ≺ ) on X = {x1,..., xn} is called a separability order if there is a collection of convex sets S and a vector −→v in some Rd such that xi ≺ x j in P(X, ≺ ) if and only if Si ≺ Sj in P(S, ≺). We prove that every partial order is the separability order of a collection of convex sets in R4, and that any poset of dimension 2 is the separability order of a set of line segments in R3. We then study the case when the convex sets are restricted to be boxes in d-dimensional spaces. We prove that any partial order is the separability order of a family of disjoint boxes in Rd for some d ≤ n 2 + 1. We prove that every poset of dimension 3 has a subdivision that is the separability order of boxes in R3, that there are partial orders of dimension 2 that cannot be realized as box separability in R3 and that for any d there are posets with dimension d that are separability orders of boxes in R3. We also prove that for any d there are partial orders with box separability dimension d; that is, d is the smallest dimension for which they are separable orders of sets of boxes in Rd