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We consider a variant of the offline Dial-a-Ride problem on weighted graphs with a single server where each request has a source and destination. The server's goal is to serve requests so as to maximize the total number of requests served within a given time limit. We first prove that no polynomial-time algorithm will always serve the optimal number of requests, even when the algorithm's time limit is augmented by any factor $ c \ge 1 $ c ≥ 1 , unless P = NP. We also show that the approximation ratio is unbounded for a reasonable class of algorithms for this problem. We then present k-Sequence, an algorithm that repeatedly serves the fastest set of k remaining requests. We show that k-Sequence has approximation ratio at most $ 2+\lceil \lambda \rceil /k $ 2 + ⌈ λ ⌉ / k and at least $ 1 + \lambda /k $ 1 + λ / k , where λ denotes the aspect ratio of the graph, and that the ratio $ 1 + \lambda /k $ 1 + λ / k is tight when $ 1 + \lambda /k \ge k $ 1 + λ / k ≥ k. We also show that even as k grows beyond the size of λ, the ratio never improves below 9/7. [ABSTRACT FROM AUTHOR]

ABSTRACT: Background: Prediction of preeclampsia in first trimester can lead to early initiation of preventative measures, as well as timely therapeutic intervention, that will prevent the maternal and perinatal morbidity and mortality. This study, prediction of preeclampsia at 11-13+6 weeks of gestation, was conducted by using serum placental growth factor (PlGF), serum pregnancy associated plasma protein – A (PAPP-A), uterine artery Doppler indices, and mean arterial pressure (MAP), in low-risk pregnant women. Methods: It is an observational longitudinal prospective study. Pregnant women with singleton pregnancies of gestational age 11-13+6 weeks were recruited. MAP, bilateral uterine artery Doppler indices, serum PAPP-A, and PlGF were measured. The follow-up of the patients was regularly done till termination of pregnancy and incidence of preeclampsia were noted. Qualitative variables were compared using Chi-square/Fisher exact test, and sensitivity and specificity of each test were analyzed. Result: Among 139 women, 27 (19%) developed preeclampsia, out of which 10 had early preeclampsia and 17 developed late preeclampsia. By combining all the parameters, the sensitivity in detecting early preeclampsia was found to be 70% and for late preeclampsia cases, 64.7%. The sensitivity of all the parameters in predicting cases of preeclampsia in general is 66.7%, specificity is 74.1%, PPV is 38.3%, and NPV is 72%. Conclusion: The maternal biomarkers serum PAPP-A, serum PlGF, uterine artery PI, MAP have sensitivity of 70% in detecting early preeclampsia cases and for late preeclampsia cases, 64.71%, in first trimester and it will help in early initiation of preventative measures as well as timely therapeutic intervention. [ABSTRACT FROM AUTHOR]