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Hierarchical Clustering: a 0.585 Revenue Approximation

Hierarchical Clustering trees have been widely accepted as a useful form of clustering data, resulting in a prevalence of adopting fields including phylogenetics, image analysis, bioinformatics and more. Recently, Dasgupta (STOC 16&#39;) initiated the analysis of these types of algorithms through the lenses of approximation. Later, the dual problem was considered by Moseley and Wang (NIPS 17&#39;) dubbing it the Revenue goal function. In this problem, given a nonnegative weight $w_{ij}$ for each pair $i,j \in [n]=\{1,2, \ldots ,n\}$, the objective is to find a tree $T$ whose set of leaves is $[n]$ that maximizes the function $\sum_{i<j \in [n]} w_{ij} (n -|T_{ij}|)$, where $|T_{ij}|$ is the number of leaves in the subtree rooted at the least common ancestor of $i$ and $j$. In our work we consider the revenue goal function and prove the following results. First, we prove the existence of a bisection (i.e., a tree of depth 2 in which the root has two children, each being a parent of $n/2$ leaves) which approximates the general optimal tree solution up to a factor of $\frac{1}{2}$ (which is tight). Second, we apply this result in order to prove a $\frac{2}{3}p$ approximation for the general revenue problem, where $p$ is defined as the approximation ratio of the Max-Uncut Bisection problem. Since $p$ is known to be at least 0.8776 (Wu et al., 2015, Austrin et al., 2016), we get a 0.585 approximation algorithm for the revenue problem. This improves a sequence of earlier results which culminated in an 0.4246-approximation guarantee (Ahmadian et al., 2019).