Hierarchical Clustering: A 0.585 Revenue Approximation

Noga Alon, Yossi Azar, Danny Vainstein

Research output: Contribution to journalConference articlepeer-review

14 Scopus citations

Abstract

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’) 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’) dubbing it the Revenue goal function. In this problem, given a nonnegative weight wij for each pair i, j ∈ [n] = {1, 2, . . ., n}, the objective is to find a tree T whose set of leaves is [n] that maximizes the function Σ i<j∈[n] wij(n− |Tij|), where |Tij| 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 1/2 (which is tight). Second, we apply this result in order to prove a 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 (Austrin et al., 2016) (Wu et al., 2015), 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).

Original languageEnglish (US)
Pages (from-to)153-162
Number of pages10
JournalProceedings of Machine Learning Research
Volume125
StatePublished - 2020
Event33rd Conference on Learning Theory, COLT 2020 - Virtual, Online, Austria
Duration: Jul 9 2020Jul 12 2020

All Science Journal Classification (ASJC) codes

  • Artificial Intelligence
  • Software
  • Control and Systems Engineering
  • Statistics and Probability

Keywords

  • Hierarchical Clustering, Approximation Algorithms

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