Correlation Clustering

We study the minimum-disagreement cluster editing problem that, from a complete edge-signed graph, where a positive edge represents two vertices agreeing to each other while a negative represents a disagreement, constructs disjoint cliques of upper bounded size such that the total number of negative edges inside cliques and positive edges outside cliques is minimized. When the sizes of the candidate cliques are unbounded, the problem is NP-hard and APX-hard, and many approximation algorithms with improving ratios have been proposed in the literature and the state-of-the-art approximation ratio of $α= 1.3865 + ε$. The capacitated variant, in which the candidate cliques have sizes no greater than a given $k$, is less studied and the best approximation ratio is $5.37$. In this paper, we study the latter problem. We first present a downsizing procedure that achieves an approximation ratio of $α_k = 2 α+ 1$, based on any $α$-approximation algorithm for the uncapacitated problem. Secondly, we propose a hybrid of cluster-LP sampling and pivoting and show that for every fractional capacitated cluster-LP solution of cost $L$, the hybrid using ordinary pivoting has an expected cost at most $\frac 95 L + \frac 1{10} \OPT_k$, and the hybrid using conditional pivoting has an expected cost at most $\frac {306}{203} L + \frac {50}{203} \OPT_k$. A cardinality-constrained adaptation of the approximate dual-separation framework of García-Soriano and Schohn supplies the required fractional solution and yields a randomized $(1.9 + ε)$-approximation algorithm and a randomized $(1.7537 + ε)$-approximation algorithm, respectively.

An increasingly important requirement for clustering algorithms is that decisions and outcomes need to be clearly understandable for human users. We address this requirement for the Correlation Clustering (CC) problem, in case the objects to be clustered are accompanied by labels. In CC one seeks to partition a set of objects into clusters based on pairwise similarity and dissimilarity information. Our proposed model only considers clusters that can be explained by a label tree, which is essentially a decision tree that partitions the objects according to labels. The goal is to find an explainable partition of the data that minimizes the CC objective function. We formalize this problem from a theoretical viewpoint, and prove several hardness of approximation results, also when constraining the size of the tree. Our proposed greedy method finds locally optimal splits, and works recursively in a top-down approach. To efficiently compute the best split, we carefully maintain certain counters. An experimental evaluation demonstrates its effectiveness. The proposed algorithm processes graphs with over million edges in a few minutes, can find the ground truth in synthetic data, and is able to produce interpretable clusters in real world networks.

Correlation Clustering (CC) is a natural formulation of clustering in combinatorial optimization, which uses a graph representation of the input and does not require a pre-specified number of clusters. Given $n$ objects and a pairwise similarity function, the goal is to cluster the objects so that similar objects are put in the same cluster and dissimilar objects are put in different clusters. Despite its versatility, existing CC algorithms suffer from significant scalability issues and are inherently transductive: i.e., the algorithm must be executed from scratch for any new problem instance. In this work, we bridge this gap by leveraging Graph Neural Networks (GNNs) to solve Inductive Correlation Clustering, a novel generalization of the CC problem designed to handle unseen graph instances. By learning to exploit common structural patterns and node features during training, our framework generalizes to new graphs drawn from the same distribution with minimal computational overhead with respect to standard algorithms. We demonstrate the effectiveness and scalability of our approach through extensive experiments. Our framework not only excels in the inductive setting, e.g., lowering the inference time up to $5$ order of magnitude, while maintaining an approximation ratio within $~10\%$ of the best baseline solution, but also achieves competitive results on standard (transductive) CC benchmarks. Finally, we showcase a practical application of our framework as a learnable pooling mechanism for graph classification. Our results indicate that our method serves as an efficient pooling layer, enhancing the ability of GNNs to capture hierarchical structural information in networks.

Correlation clustering is a fundamental unsupervised learning problem. On complete graphs, both the min-disagreement and min-max objectives admit constant-factor approximations, yet on general (non-complete) graphs, the best guarantees blow up to $O(\log n)$ and $O(\sqrt{n})$. This gap between the two regimes motivates the following question: are there classes of incomplete graphs that circumvent the lower bounds on general graphs and admit approximation guarantees approaching those attainable on complete graphs? We study a natural class of graphs obtained by randomly subsampling a complete signed graph $G$, where each edge is independently deleted with probability $q$. For such graph instances both for the min-max and the min-disagreement objectives, we prove approximation guarantees (depending on $q$) that are substantially better than the bounds achievable for general graphs. We supplement our theoretical results with experiments that also suggest that the approximation ratios of our algorithm are close to those of the complete graph and better than the worst-case bounds for general (non-complete) graphs.

Clustered federated learning benefits from organizing heterogeneous participants into coalitions that train coalition-specific models, but such clustering is sustainable only if participants prefer their assigned coalition and the required transfers are affordable. We develop a transferable-surplus model separating learning benefit, system cost, participant cost, and monetary transfers; an allocation rule converts coalition surplus into hedonic preferences, and weak budget feasibility guarantees nonnegative retained coordinator surplus. For symmetric pairwise allocations the induced game is an exact potential game: a Nash-stable partition exists, every strict better-response process converges, and with destination consent accepted better responses reach an individually stable partition. We characterize feasibility of bounded pair incentives and verify the exponentially many budget constraints in polynomial oracle time when retained slack is submodular. Decomposing welfare into participant potential and retained slack yields additive and multiplicative price-of-stability guarantees, the latter asymptotically tight; exact balance gives welfare-optimal stability only on the pairwise-representable class, and budget feasibility alone permits unbounded welfare loss. Global potential maximization equals weighted maximum-agreement correlation clustering, and approximation followed by stabilization satisfies an end-to-end welfare bound governed by retained slack and negative-edge mass, attained by an explicit construction. In a preregistered five-seed CIFAR-10 study the mechanism reaches the certified estimated-table welfare optimum on every primary instance, equal-surplus sharing has no Nash-stable outcome on three, and pairwise validation gain gives far more reliable pair signs than gradient alignment.

Understanding activities of Internet scanners is challenging; it often requires identifying relationships between sources, a task for which semantic annotations are scarce. This work investigates whether semantically meaningful pairwise relationships between sequences of network flow records can be estimated by contrastive learning, without pretraining and without annotations. To this end, we propose a transformer model that embeds minimally preprocessed sequences of network flow records and train it using contrastive learning. With the similarities obtained from this model, we state a correlation clustering problem and solve it locally. Experimentally, we show: Learned similarities are higher on average for sequences originating from the same source than for sequences originating from different sources, and this property generalizes to unseen sequences of unseen sources. Moreover, correlation clustering yields clusters consistent with scanner labels. The complete source code of the algorithms and for reproducing the experiments is publicly available.

Chromatic Correlation Clustering (CCC) extends Correlation Clustering by assigning semantic colors to edges and requiring each cluster to receive a single color label. Unlike standard CC, whose LP relaxation has integrality gap 2 on complete graphs and admits a 2.06-approximation, the analogous LP for CCC has a strict lower bound of 2.11, and the best known LP-rounding algorithm achieves 2.15. We explain this gap by isolating the source of difficulty: cross-edge chromatic interference. Neutral edges, whose color does not match the candidate cluster color, create an irreducible cost absent from standard CC and force any color-independent rounding scheme to pay an additional mismatch penalty. We make four contributions. First, we prove a Global Integrality Gap Decomposition Theorem showing that the gap of any color-independent CCC rounding algorithm equals the standard CC gap plus an irreducible chromatic penalty Delta(L) > 0. Second, we solve the associated min-max problem and derive the staircase formula Delta(L) = ((L-1)/L) Delta_infinity, where Delta_infinity is approximately 0.0734. In particular, the two-color gap is 2.0967, separating CCC from standard CC already at L = 2. Third, we introduce Color-Coupled Correlation Clustering (C4). Adding the valid global constraint sum_c x_uv^c >= L-1 and a correlated interval-packing rounding scheme makes neutral edges behave like classical negative edges, recovering the optimal 2.06 approximation and bypassing the 2.11 lower bound for the uncoupled LP. Fourth, experiments on extremal instances, real multi-relational networks, and fairness benchmarks validate the theory: empirical LP gaps follow the predicted staircase, and C4 matches the unconstrained approximation ratio under fairness constraints.

This work aims to improve the sample efficiency of parallel large-scale ranking and selection (R&S) problems by leveraging correlation information. We modify the commonly used "divide and conquer" framework in parallel computing by adding a correlation-based clustering step, transforming it into "clustering and conquer". Theoretically, we develop a novel gradient-based analysis framework and show that this seemingly simple modification substantially improves the performance of large-scale R&S procedures. Our approach enjoys two key advantages: (1) it does not require highly accurate correlation estimation or precise clustering, and (2) it can be seamlessly integrated with various existing fixed-precision and fixed-budget R&S procedures while achieving optimal sample complexity. We also introduce a new parallel clustering algorithm tailored to large-scale settings. Finally, in large-scale AI applications such as neural architecture search, our methods demonstrate superior performance.