A Tight Bound for Shortest Augmenting Paths on Trees

The shortest augmenting path technique is one of the fundamental ideas used in maximum matching and maximum flow algorithms. Since being introduced by Edmonds and Karp in 1972, it has been widely applied in many different settings. Surprisingly, despite this extensive usage, it is still not well understood even in the simplest case: online bipartite matching problem on trees. In this problem a bipartite tree $T=(W \uplus B, E)$ is being revealed online, i.e., in each round one vertex from $B$ with its incident edges arrives. It was conjectured by Chaudhuri et. al. [K. Chaudhuri, C. Daskalakis, R. D. Kleinberg, and H. Lin. Online bipartite perfect matching with augmentations. In INFOCOM 2009] that the total length of all shortest augmenting paths found is $O(n \log n)$. In this paper, we prove a tight $O(n \log n)$ upper bound for the total length of shortest augmenting paths for trees improving over $O(n \log^2 n)$ bound [B. Bosek, D. Leniowski, P. Sankowski, and A. Zych. Shortest augmenting paths for online matchings on trees. In WAOA 2015].Comment: 22 pages, 10 figure

Shortest augmenting paths for online matchings on trees

The shortest augmenting path (Sap) algorithm is one of the most classical approaches to the maximum matching and maximum flow problems, e.g., using it Edmonds and Karp (J. ACM 19(2), 248–264 1972) have shown the first strongly polynomial time algorithm for the maximum flow problem. Quite astonishingly, although it has been studied for many years already, this approach is far from being fully understood. This is exemplified by the online bipartite matching problem. In this problem a bipartite graph G = (W ⊎ B, E) is being revealed online, i.e., in each round one vertex from B with its incident edges arrives. After arrival of this vertex we augment the current matching by using shortest augmenting path. It was conjectured by Chaudhuri et al. (INFOCOM’09) that the total length of all augmenting paths found by Sap is O(nlogn). However, no better bound than O(n2) is known even for trees. In this paper we prove an O(nlog2n) upper bound for the total length of augmenting paths for trees

Saliva as Blood Alternative in Therapeutic Monitoring of Teriflunomide—Development and Validation of the Novel Analytical Method

Therapeutic drug monitoring (TDM) is extremely helpful in individualizing dosage regimen of drugs with narrow therapeutic ranges. It may also be beneficial in the case of drugs characterized by serious side effects and marked interpatient pharmacokinetic variability observed with leflunomide and its biologically active metabolite, teriflunomide. One of the most popular matrices used for TDM is blood. A more readily accessible body fluid is saliva, which can be collected in a much safer way comparing to blood. This makes it especially advantageous alternative to blood during life-threatening SARS-CoV-2 pandemic. However, drug’s saliva concentration is not always a good representation of its blood concentration. The aim of this study was to verify whether saliva can be used in TDM of teriflunomide. We also developed and validated the first reliable and robust LC-MS/MS method for quantification of teriflunomide in saliva. Additionally, the effect of salivary flow and swab absorptive material from the collector device on teriflunomide concentration in saliva was evaluated. Good linear correlation was obtained between the concentration of teriflunomide in plasma and resting saliva (p p p = 0.198) or type of swab in the Salivette device on saliva’s teriflunomide concentration was detected. However, to reduce variability the use of stimulated saliva and synthetic swabs is advised