Convexity in partial cubes: the hull number

We prove that the combinatorial optimization problem of determining the hull number of a partial cube is NP-complete. This makes partial cubes the minimal graph class for which NP-completeness of this problem is known and improves some earlier results in the literature. On the other hand we provide a polynomial-time algorithm to determine the hull number of planar partial cube quadrangulations. Instances of the hull number problem for partial cubes described include poset dimension and hitting sets for interiors of curves in the plane. To obtain the above results, we investigate convexity in partial cubes and characterize these graphs in terms of their lattice of convex subgraphs, improving a theorem of Handa. Furthermore we provide a topological representation theorem for planar partial cubes, generalizing a result of Fukuda and Handa about rank three oriented matroids.Comment: 19 pages, 4 figure

Open questions in utility theory

Throughout this paper, our main idea is to explore different classical questions arising in Utility Theory, with a particular attention to those that lean on numerical representations of preference orderings. We intend to present a survey of open questions in that discipline, also showing the state-of-art of the corresponding literature.This work is partially supported by the research projects ECO2015-65031-R, MTM2015-63608-P (MINECO/ AEI-FEDER, UE), and TIN2016-77356-P (MINECO/ AEI-FEDER, UE)

Many vulnerable or a few resilient specimens? Finding the optimal for reintroduction/restocking programs

Most reintroduction and restocking programs consist of releasing captive-raised juveniles. The usefulness of these programs has been questioned, and therefore, quality control is advisable. However, evaluating restocking effectiveness is challenging because mortality estimation is required. Most methods for estimating mortality are based on tag recovery. In the case of fish, juveniles are tagged before release, and fishermen typically recover tags when fish are captured. The statistical models currently available for analyzing these data assume either constant mortality rates, fixed tag non-reporting rates, or both. Here, instead, we proposed a method that considers the mortality rate variability as a function of age/size of the released juveniles. Furthermore, the proposed method can disentangle natural from fishing mortality, analyzing the temporal distribution of the captures reported by fishermen from multiple release events. This method is demonstrated with a restocking program of a top-predator marine fish, the meagre (Argyrosomus regius), in the Balearic Islands. The estimated natural mortality just after release was very high for young fish (m = 0.126 day for fish 180 days old), but it was close to zero for large/old fish. These large/old fish were more resilient to wild conditions, although a long time was needed to achieve a relevant reduction in natural mortality. Conversely, these large/old fish were more vulnerable to fishing, creating a trade-off in survival. The release age that maximizes the number of survivors after, for example, one year at liberty was estimated to be 1,173 days. However, the production cost of relatively old fish is high, and only a few fish can be produced and released within a realistic budget. Therefore, in the case of the meagre, increasing the number of released fish will have no or scarce effects on restocking success. Conversely, it is advisable implement measures to reduce the high natural mortality of young juveniles and/or the length of time needed to improve fish resilience.This work was financially supported by >Instituto Nacional de Investigación y Tecnología Agraria y Alimentaria> (INIA) belonging to >Ministerio de Economía y Competitividad> (Spanish Government), through the projects RTA-2007 00033-C02-01 and RTA-2011 00056-C02-00Peer Reviewe

Chain Representations of Nested Families of Biorders

International audienc