Some Open Problems in Parameterized Complexity Related to the Work of Jianer Chen

This short paper highlights some open problems related to the work of Jianer Chen in the area of parameterized/multivariate algorithmics.     

Dynamic Dominating Set and Turbo-Charging Greedy Heuristics

The main purpose of this paper is to exposit two very different, but very general, motivational schemes in the art of parameterization and a concrete example connecting them. We introduce a dynamic version of the DOMINATING SET problem and prove that it is fixed-parameter tractable（FPT）. The problem is motivated by settings where problem instances evolve. It also arises in the quest to improve a natural greedy heuristic for the DOMINATING SET problem.     

Arboreality predicts Batrachochytrium dendrobatidis infection level in tropical direct-developing frogs

Pathogen-mediated changes in host behaviour can result from hosts altering their habitat preferences. Although infection risk with pathogenic fungus Batrachochytrium dendrobatidis in amphibians is associated with environments favouring its growth, the relationship with microhabitat use has not been examined. Here, we aim to determine if microhabitats used by frogs during their nocturnal activity predict B. dendrobatidis prevalence and infection intensity. Our focal host, Eleutherodactylus coqui, is a habitat generalist that uses multiple habitats from the forest floor to the canopy. We analysed data on B. dendrobatidis occurrence in 157 adults and 122 juveniles at El Yunque National forest in Puerto Rico. We categorized each individual’s nocturnal microhabitat as forest floor, curled palm fronds in the floor, arboreal bromeliads and foliage or tree trunks 50 cm to 2.5 m above ground. We found that frogs on the forest floor had the greatest B. dendrobatidis prevalence (73%), compared with those active in vegetation above ground (55%). Overall, the probability of B. dendrobatidis infection in frogs using microhabitats on the forest floor was twice as great as for those on arboreal substrates. Differences in B. dendrobatidis prevalence and intensity in E. coqui may be explained by specific abiotic conditions of microenvironments (temperature and humidity) affecting both pathogen and host, and by the age-specific ecological requirements of hosts. Adults were found to be most active in microhabitats where individuals had lower infection burdens, suggesting pathogen-modulated habitat choice. This work has important implications for the evolutionary dynamics of enzootic diseases and provides data that may inform potential mitigation strategies against a generalist amphibian pathogen.     

Spawning behaviour of Engystomops pustulatus (Anura: Leptodactylidae)

Foam nests have evolved independently in several amphibian groups as an adaptive response to prevent predation and desiccation in dry environments. Nests are normally laid on ponds, or in underground galleries, humid forest leaf litter or terrestrial bromeliads. They are built when males or females beat a foam precursor associated with the egg masses extruded by the female. The spawning process requires the synchronic actions of the mating pair to obtain a hemispheric nest that protects the offspring. Herein, we describe the spawning behaviour of Engystomops pustulatus based on videos from 13 nesting couples from the lowlands of western Ecuador. Three variables were measured as indicators of male effort: duration of mixing events, duration of resting periods, and number of kicks per mixing event. We consider that not only male physical effort but also female behaviour influences nest structure. We suggest that nest building requires prolonged and intense physical activity by the male as well as the female’s steady position during spawning and female’s oviposition site selection. Nest building has two phases. In the first phase, the duration of resting periods, the duration of mixing events, and the number of kicks increase and are highly variable. During the second phase the three variables stabilise until the end. The volume of the nest increased mainly during the second phase. In four nesting events we observed kicking movements by the female. To our knowledge, this is the first time that female kicking has been observed in leptodactylid frogs. The function of this behaviour is unknown but our observations suggest that it may be triggered by insufficient male effort. Traditionally, female mate choice in Engystomops has been explained under models of indirect benefits exclusively. We argue that the prolonged male activity during nesting could influence female fitness directly. This will allow the operation of sexual selection via direct benefits.