Online Node- and Edge-Deletion Problems with Advice

In online edge- and node-deletion problems the input arrives node by node and an algorithm has to delete nodes or edges in order to keep the input graph in a given graph class $$\Pi $$ Π at all times. We consider only hereditary properties $$\Pi $$ Π , for which optimal online algorithms exist and which can be characterized by a set of forbidden subgraphs $${{\mathcal{F}}}$$ F and analyze the advice complexity of getting an optimal solution. We give almost tight bounds on the Delayed Connected$${{\mathcal{F}}}$$ F -Node-Deletion Problem, where all graphs of the family $${\mathcal{F}}$$ F have to be connected and almost tight lower and upper bounds for the Delayed$$H$$ H -Node-Deletion Problem, where there is one forbidden induced subgraph H that may be connected or not. For the Delayed$$H$$ H -Node-Deletion Problem the advice complexity is basically an easy function of the size of the biggest component in H. Additionally, we give tight bounds on the Delayed Connected$${\mathcal{F}}$$ F -Edge-Deletion Problem, where we have an arbitrary number of forbidden connected graphs. For the latter result we present an algorithm that computes the advice complexity directly from $${\mathcal{F}}$$ F . We give a separate analysis for the Delayed Connected$$H$$ H -Edge-Deletion Problem, which is less general but admits a bound that is easier to compute.

Profile and complexity of travel medicine consultations in Chile: unicentric cross-sectional study

ObjectiveTo analyse the spectrum, vaccination needs and pretravel advice complexity of travellers presenting at a travel medicine clinic in Santiago, Chile.DesignCross-sectional study.SettingPretravel consultations in a private healthcare centre in Chile, an ‘emerging market’ country in South America.ParticipantsTravellers (n=1341) seeking pretravel advice at the Travel Medicine Program of Clínica Alemana, Santiago, from April 2016 to March 2018.Primary and secondary outcome measuresDemographical and travel characteristics, indications for travel vaccines and malaria prophylaxis, and complexity of travel consultations.ResultsOf 1341 travellers, 51% were female; the median age was 33 years. Most frequent travel reasons were tourism (67%) and business (20%). Median travel duration and time to departure were 21 days and 28 days, respectively. Most destinations were located in America (41%), followed by Asia (36%) and Africa (26%); 96% visited less developed countries, mostly in tropical regions, with risk of arboviral infections (94%) and malaria (69%). The indicated vaccine indications comprised hepatitis A (84%), yellow fever (58%), typhoid fever (51%), rabies (29%), polio (8%), Japanese encephalitis (6%) and meningococcal meningitis (5%). More than 60% of consultations were classified as complex.ConclusionThe studied population mostly visited less developed tropical regions, resulting in a high requirement of yellow fever and other travel-related vaccinations. Most consultations were complex and required a comprehensive knowledge and training in travel medicine.

Online Minimum Spanning Tree with Advice

In the online minimum spanning tree problem, a graph is revealed vertex by vertex; together with every vertex, all edges to vertices that are already known are given, and an online algorithm must irrevocably choose a subset of them as a part of its solution. The advice complexity of an online problem is a means to quantify the information that needs to be extracted from the input to achieve good results. For a graph of size [Formula: see text], we show an asymptotically tight bound of [Formula: see text] on the number of advice bits to produce an optimal solution for any given graph. For particular graph classes, e.g., with bounded degree or a restricted edge weight function, we prove that the upper bound can be drastically reduced; e.g., [Formula: see text] advice bits allow to compute an optimal result if the weight function equals the Euclidean distance; if the graph is complete and has two different edge weights, even a logarithmic number suffices. Some of these results make use of the optimality of Kruskal’s algorithm for the offline setting. We also study the trade-off between the number of advice bits and the achievable competitive ratio. To this end, we perform a reduction from another online problem to obtain a linear lower bound on the advice complexity for any near-optimal solution. Using our results finally allows us to give a lower bound on the expected competitive ratio of any randomized online algorithm for the problem, even on graphs with three different edge weights.

Online Matching in Regular Bipartite Graphs

In an online problem, the input is revealed one piece at a time. In every time step, the online algorithm has to produce a part of the output, based on the partial knowledge of the input. Such decisions are irrevocable, and thus online algorithms usually lead to nonoptimal solutions. The impact of the partial knowledge depends strongly on the problem. If the algorithm is allowed to read binary information about the future, the amount of bits read that allow the algorithm to solve the problem optimally is the so-called advice complexity. The quality of an online algorithm is measured by its competitive ratio, which compares its performance to that of an optimal offline algorithm. In this paper we study online bipartite matchings focusing on the particular case of bipartite matchings in regular graphs. We give tight upper and lower bounds on the competitive ratio of the online deterministic bipartite matching problem. The competitive ratio turns out to be asymptotically equal to the known randomized competitive ratio. Afterwards, we present an upper and lower bound for the advice complexity of the online deterministic bipartite matching problem.