TL;DR
A randomized incremental ( 𝑘, 𝑧 ) -clustering algorithm that maintains with high probability a constant-factor approximation in a graph undergoing edge insertions with a total update time of ˜ 𝑂 ( 𝑘𝑚 1 + 𝑜 ( 1 ) + 𝑘 1 + 1 𝜆 𝑚 ) , where 𝜆 ≥ 1 is an arbitrary fixed constant.
This work identifies Constrained Correlation Clustering as a variant of Correlation Clustering for which optimal approximations might be within reach, and significantly tighten the gap, by showing that the optimal approximation factor lies in $[2,\frac{16}{7}-\gamma)$ for a small constant $0$.
It is proved that the problem of defining a bounded cost set of nodes S such that the influence spreading from S in G, within a given time bound, is as large as possible, and that the problem is NP-hard, even in simple networks like complete graphs and trees.
This work exponentially improves upon the approximation ratio of an existing algo-rithm to 1 r +1, presents another novel dependent-rounding algorithm with an approximation ratio of 1 / ⌈ k 2 ⌉, and modify the initial algorithm via analytical scaling techniques to achieve an approximation factor of (1 − e − r ) /r.
An online algorithm based on a peeling strategy to compute (k,l,η)-cores and a lightweight DUCS index, which stores directional probability information separately, reducing storage overhead while still pruning many irrelevant vertices, is presented.
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