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 $0$.
Abstract
In the Correlation Clustering problem, we are given an undirected graph and are tasked with computing a clustering (partition of the nodes) that minimizes the number of violated pairs (edges across different clusters plus non-edges within clusters). In the constrained version of this problem, the goal is to compute a clustering that satisfies additional hard constraints mandating certain pairs to be in the same cluster and certain pairs to be in different clusters. In this work, we identify Constrained Correlation Clustering as a variant of Correlation Clustering for which optimal approximations might be within reach, and make progress towards this front. Constrained Correlation Clustering is APX-Hard, and the optimal approximation factor is known to lie in $(\frac{24}{23},3]$. We significantly tighten this gap, by showing that the optimal approximation factor lies in $[2,\frac{16}{7}-\gamma)$ for a small constant $\gamma>0$. Our lower bound of $2$ shows a separation between Correlation Clustering (which admits an $1.485+\epsilon$ approximation) and Constrained Correlation Clustering\footnote{The same hardness result was obtained independently by Cao and Xu~\cite{cao2026clusterdeletionhardapproximate}.}. Our upper bound of $\frac{16}{7}-\gamma$ uses the Sherali-Adams relaxation and goes beyond straightforward Triangle-Based analysis; more precisely, our algorithm belongs to a natural class of pivoting algorithms for which we prove that a straightforward Triangle-Based analysis cannot prove a better-than-$\frac{16}{7}$ approximation. Finally, as a byproduct of our techniques, we completely resolve the approximability of Cluster Deletion. Cluster Deletion is a well-studied special case of Constrained Correlation Clustering for which a $2$-approximation algorithm is known. We show that this is optimal, as our lower bound holds even for this special case.
For graph instances both for the min-max and the min-disagreement objectives, this work proves approximation guarantees that are substantially better than the bounds achievable for general graphs.
N. RajathRaoK., Jens Schlöter, Sami Davies et al.· 0 citations
We study the \emph{connected $k$-median} clustering problem, a clustering problem that augments the classical $k$-median objective with connectivity constraints. We focus on the \emph{overlapping} variant of the problem, where clusters are allowed to share vertices. In addition to a metric space $(V,d)$, the input contains a connected graph $G$ on the same vertex set $V$ of size $n$. The goal is to select at most $k$ centers $C$ and assign vertices to them so as to minimize the $k$-median cost (i.e., $\sum_{v\in V} d(v,C)$), subject to the constraint that each cluster induces a connected subgraph of $G$. Since the metric space and the connectivity graph are independent, the problem is significantly more challenging than standard clustering. Eube et al.~\cite{eube2025esa} showed that even the assignment version is $\Omega(\log n)$-hard to approximate and gave approximation algorithms with guarantees depending polynomially on $k$. We develop a configuration-LP-based framework that combines covering LP techniques with a rooted minimum-density oracle. For the assignment version, we obtain an $O(\log^2 n)$-approximation. For the general version, we develop a bicriteria framework that opens $O(k\log n)$ centers while achieving an $O(\log^2 n)$-approximation in cost. %Our results provide a different LP-based approach for handling connectivity constraints in clustering problems and demonstrate that configuration LPs, covering LPs, and rooted density oracles can be combined effectively to obtain approximation guarantees for clustering objectives under graph-theoretic constraints.
Kushagra Chatterjee, Rojin Rezvan, A. Vakilian· 0 citations
This work uses Graph Neural Networks (GNNs) to solve Inductive Correlation Clustering, a novel generalization of the CC problem designed to handle unseen graph instances, and indicates that the method serves as an efficient pooling layer, enhancing the ability of GNNs to capture hierarchical structural information in networks.
Francesco Paolo Nerini, Francesco Bonchi, Arijit Khan et al.· 0 citations
Many existing clustering methods are designed based on a set-oriented definition---a cluster is a set of similar points---relying a point-to-point similarity function to find similar points. This works well for compact clusters, but clustering performance can deteriorate badly when cluster shapes are irregular, and densities or sizes vary between clusters. Recent `Cluster-as-Distribution'(CaD) clustering has been shown to discover these generic types of clusters in practice by treating each cluster as a set of independent and identically distributed points generated from some unknown distribution via a greedy search, achieving a clustering objective equivalent to that of Spectral Clustering, but with better clustering outcomes without eigen-decomposition. However, a theoretical analysis of this phenomenon is still lacking. Our analyses are from two angles. First, we analyze the approximation error between the true and empirical distribution embeddings. Second, we show that the greedy search employed to achieve the CaD clustering objective can be mapped to a partition matroid---yielding greedy optimality. These yield a near-optimality guarantee for the CaD clustering objective, with regret controlled by the approximation error. This is the first analysis that explains why CaD clustering via greedy search can discover clusters of arbitrary shapes, densities and sizes (where all set-oriented clustering methods have failed to discover) when the estimated cluster embeddings faithfully approximate the underlying cluster distributions.
Kaifeng Zhang, Kai-Ming Ting, S. Chawla· arXiv.org· 0 citations
The cost of a hierarchical clustering can be represented by an ultrametric whose lowest-common-ancestor labels are cluster cardinalities. We relate this known representation to the shortest-path geometry of a similarity graph. For a connected support graph $G$, let $d_G$ be its unit-length shortest-path metric and let the edge weights enter only the objective. We prove that the shifted Dasgupta optimum is exactly the minimum edge-weighted cost of a cardinality-realizable ultrametric that dominates $d_G$. Connectedification lemmas put this problem and its freely labeled dominating-ultrametric relaxation on the same class of connected binary hierarchies, labeled respectively by cardinality and graph diameter. As a sharp baseline, we determine the exact worst-case price of cardinality realizability: on every $n$-vertex instance the ratio of the two optima is at most $(2n-1)/3$, with equality on the unweighted complete graph; the sharp factor for the standard unshifted objective is $2(n+1)/3$. Our principal structural result bounds this gap by a hereditary weighted fragmentation profile defined through connected balanced cuts. Uniform local control gives an $O(\log n)$ gap, polynomial decay gives a constant gap, and the logarithmic order is tight even for unweighted trees of maximum degree $3$. On locally regular bounded-degree trees, the hierarchy can be constructed in $O(n\log n)$ time. An energy decomposition and a geometric density bound provide supporting instance-sensitive estimates. Thus the cardinality label has an unavoidable linear worst case but admits substantially smaller bounds on natural sparse graph classes.
By using a simple textbook reduction from vertex cover, we show that the following three problems are all UG-hard to approximate with constant-factor smaller than two; minimum weakness strong triadic closure, cluster deletion and constrained correlation clustering. Additionally, we analyze the well-known low-diameter decomposition by Calinescu, Karloff and Raban applied to the standard LP relaxation semi-metric for constrained correlation clustering. As opposed to traditional pivot-based approaches, a CKR partition elegantly handles must-link and cannot-link constraints. It guarantees a 3-approximation in expectation, which matches the original approximation ratio by van Zuylen and Williamson. We conjecture that it in fact achieves a strictly better than 3-approximation, yet this remains an open problem.
Florian Adriaens· 0 citations
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