SUMMARY BACKGROUND Physiotherapy services are necessary for hospitalized patients of COVID-19 as well as chronic patients. Thus, physiotherapists present an increased risk of exposure to coronavirus. This study aimed to determine the number of physiotherapists who interrupted their services because of the COVID-19 pandemic and to verify the procedures adopted by the ones who are still working. METHODS The sample comprised 619 physiotherapists who worked in Portugal, 154 (24.9%) male and 465 (75.1%) female, aged between 22 and 67 years (34.47±8.70). The measurement instrument was an on-line questionnaire applied in late March 2020 through contacts and social networks. RESULTS 453 (73.2%) physiotherapists interrupted their work activities in person because of the pandemic and 166 (26.8%) continue to work in person. The main measures adopted by physical therapists who continue to work in person included: hand washing (21.5%), mask use (20.3%), material disinfection (19.3%) and, glove use (19.3%). Of the physiotherapists who are not working in person (n = 453), 267 (58.9%) continue to monitor their patients at a distance, and 186 (41.1%) are not monitoring the patients. The main measures used by physiotherapists to monitor their patients at a distance included: written treatment prescription (38%), making explanatory videos (26.7%), and synchronous video conference treatment (23.5%). CONCLUSIONS Our data revealed that most of the physiotherapists interrupted their face-to-face practices because of the COVID-19 pandemic, however, once they do not follow up their patients’ treatment in person, most of them adapted to monitor their patients from a distance.

Abstract. The attainment function provides a description of the location of the distribution of a random non-dominated point set. This function can be estimated from experimental data via its empirical counterpart, the empirical attainment function (EAF). However, computation of the EAF in more than two dimensions is a non-trivial task. In this article, the problem of computing the empirical attainment function is formalised, and upper and lower bounds on the corresponding number of output points are presented. In addition, efficient algorithms for the two and three-dimensional cases are proposed, and their time complexities are related to lower bounds derived for each case.

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Given a nondominated point set [Formula: see text] of size [Formula: see text] and a suitable reference point [Formula: see text], the Hypervolume Subset Selection Problem (HSSP) consists of finding a subset of size [Formula: see text] that maximizes the hypervolume indicator. It arises in connection with multiobjective selection and archiving strategies, as well as Pareto-front approximation postprocessing for visualization and/or interaction with a decision maker. Efficient algorithms to solve the HSSP are available only for the 2-dimensional case, achieving a time complexity of [Formula: see text]. In contrast, the best upper bound available for [Formula: see text] is [Formula: see text]. Since the hypervolume indicator is a monotone submodular function, the HSSP can be approximated to a factor of [Formula: see text] using a greedy strategy. In this article, greedy [Formula: see text]-time algorithms for the HSSP in 2 and 3 dimensions are proposed, matching the complexity of current exact algorithms for the 2-dimensional case, and considerably improving upon recent complexity results for this approximation problem.

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Given a non-dominated point set X ⊂ R d of size n and a suitable reference point r ∈ R d , the Hypervolume Subset Selection Problem (HSSP) consists of finding a subset of size k ≤ n that maximizes the hypervolume indicator. It arises in connection with multiobjective selection and archiving strategies, as well as Pareto-front approximation postprocessing for visualization and/or interaction with a decision maker. Efficient algorithms to solve the HSSP are available only for the 2-dimensional case, achieving a time complexity of O(n(k + log n)). In contrast, the best upper bound available for d > 2 is O(n d 2 log n + n n−k ). Since the hypervolume indicator is a monotone submodular function, the HSSP can be approximated to a factor of (1 − 1/e) using a greedy strategy. Such a greedy algorithm for the 3-dimensional HSSP is proposed in this paper. The time complexity of the algorithm is shown to be O(n 2 ), which considerably improves upon recent complexity results for this approximation problem.