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This study considers the soft capacitated vertex cover problem in a dynamic setting. This problem generalizes the dynamic model of the vertex cover problem, which has been intensively studied in recent years. Given a dynamically changing vertex-weighted graph G = (V , E) , which allows edge insertions and edge deletions, the goal is to design a data structure that maintains an approximate minimum vertex cover while satisfying the capacity constraint of each vertex. That is, when picking a copy of a vertex v in the cover, the number of v's incident edges covered by the copy is up to a given capacity of v. We extend Bhattacharya et al.'s work [SODA'15 and ICALP'15] to obtain a deterministic primal-dual algorithm for maintaining a constant-factor approximate minimum capacitated vertex cover with O (log n / ϵ) amortized update time, where n is the number of vertices in the graph. The algorithm can be extended to (1) a more general model in which each edge is associated with a non-uniform and unsplittable demand, and (2) the more general capacitated set cover problem. [ABSTRACT FROM AUTHOR]

The subtree transfer (STT) distance is one of the distance metric for comparing phylogenies. Previous work on computing the STT distance considered degree-3 trees only. In this paper, we give an approximation algorithm for the STT distance for degree-d trees with arbitrary d and with generalized STT operations. Also, some NP-hardness results related to STT distance are presented. [ABSTRACT FROM AUTHOR]