Technology Category
Constraint Satisfaction and Optimization: Distributed CSP/OptimizationMachine Learning: ClusteringSearch and Optimization: Distributed Search
Handling LP-Rounding for Hierarchical Clustering and Fitting Distances by Ultrametrics
🤖 AI Summary
This paper studies the multi-level hierarchical correlation clustering problem: given a complete graph and multi-layer input information, one must compute nested vertex partitions—each finer partition refining the coarser one above—so as to minimize the weighted number of inconsistent edges across all layers. Equivalently, this amounts to fitting a distance matrix with an ultrametric. Addressing the limitations of prior approaches—namely, large approximation ratios and inadequate modeling of hierarchical structure—we propose a novel LP-rounding paradigm. We reveal that hierarchical clustering fundamentally reduces to finding cuts satisfying average-distance constraints, and unify ultrametric violation distances within this framework. By designing a tailored LP relaxation and a rounding scheme leveraging graph cuts and average-distance structural properties, we achieve an approximation ratio of 25.7846—substantially improving upon the best known results since 2021—and derive a concise new algorithm for ultrametric fitting.
Technology Category
Application Category
📝 Abstract
We consider the classic correlation clustering problem in the hierarchical setting. Given a complete graph $G=(V,E)$ and $ell$ layers of input information, where the input of each layer consists of a nonnegative weight and a labeling of the edges with either + or -, this problem seeks to compute for each layer a partition of $V$ such that the partition for any non-top layer subdivides the partition in the upper-layer and the weighted number of disagreements over the layers is minimized. Hierarchical correlation clustering is a natural formulation of the classic problem of fitting distances by ultrametrics, which is further known as numerical taxonomy in the literature. While single-layer correlation clustering received wide attention since it was introduced and major progress evolved in the past three years, few is known for this problem in the hierarchical setting. The lack of understanding and adequate tools is reflected in the large approximation ratio known for this problem originating from 2021. In this work we make both conceptual and technical contributions towards the hierarchical clustering problem. We present a simple paradigm that greatly facilitates LP-rounding in hierarchical clustering, illustrated with an algorithm providing a significantly improved approximation guarantee of 25.7846 for the hierarchical correlation clustering problem. Our techniques reveal surprising new properties of the formulation presented and subsequently used in previous works for hierarchical clustering over the past two decades. This provides an interpretation on the core problem in hierarchical clustering as the problem of finding cuts with prescribed properties regarding average distances. We further illustrate this perspective by showing that a direct application of the techniques gives a simple alternative to the state-of-the-art result for the ultrametric violation distance problem.
Problem
Research questions and friction points this paper is trying to address.
Hierarchical correlation clustering with multiple edge-labeled layers
Minimizing weighted disagreements in hierarchical partitions
Improving LP-rounding for ultrametric distance fitting
Innovation
Methods, ideas, or system contributions that make the work stand out.
LP-rounding paradigm for hierarchical clustering
Improved approximation guarantee 25.7846
Interprets core problem as cut-finding
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