Title: Improved Approximation Algorithms for Chromatic and Pseudometric-Weighted Correlation Clustering
Venue: NeurIPS 2025
Accepted: yes
Keywords: Correlation Clustering

Chromatic Clustering

Approximation Algorithms

Graph Algorithms
Abstract: Correlation Clustering (CC) is a foundational problem in unsupervised learning that models binary similarity relations using labeled graphs. While classical CC has been well studied, many real-world applications involve more nuanced relationships—either multi-class categorical interactions or varying confidence levels in edge labels. To address these, two natural generalizations have been proposed: Chromatic Correlation Clustering (CCC), which assigns semantic colors to edge labels, and pseudometric-weighted CC, which allows edge weights satisfying the triangle inequality. In this paper, we develop improved approximation algorithms for both settings. Our approach leverages LP-based pivoting techniques combined with problem-specific rounding functions. For the pseudometric-weighted correlation clustering problem, we present a tight $\frac{10}{3}$-approximation algorithm, matching the best possible bound achievable within the framework of standard LP relaxation combined with specialized rounding. For the Chromatic Correlation Clustering (CCC) problem, we improve the approximation ratio from the previous best of $2.5$ to  $2.15$, and we establish a lower bound of $2.11$ within the same analytical framework, highlighting the near-optimality of our result.

Improved Approximation Algorithms for Chromatic and Pseudometric-Weighted Correlation Clustering

Abstract

Correlation Clustering (CC) is a foundational problem in unsupervised learning that models binary similarity relations using labeled graphs. While classical CC has been widely studied, many real-world applications involve more nuanced relationships, either multi-class categorical interactions or varying confidence levels in edge labels. To address these, two natural generalizations have been proposed: Chromatic Correlation Clustering , which assigns semantic colors to edge labels, and pseudometric-weighted Correlation Clustering , which allows edge weights satisfying the triangle inequality. In this paper, we develop improved approximation algorithms for both settings. Our approach leverages LP-based pivoting techniques combined with problem-specific rounding functions. For the pseudometric-weighted correlation clustering problem, we present a tight 10 3 -approximation algorithm, matching the best possible bound achievable within the framework of standard LP relaxation combined with specialized rounding. For the Chromatic Correlation Clustering (CCC) problem, we improve the approximation ratio from the previous best of 2 . 5 to 2 . 15 , and we establish a lower bound of 2 . 11 within the same analytical framework, highlighting the near-optimality of our result.

1 Introduction

Clustering is a fundamental task in unsupervised learning, where the goal is to partition a set of objects into groups based on their pairwise relationships. One prominent problem in this domain is Correlation Clustering (CC) [5], which models binary similarity/dissimilarity between items using an edge-labeled graph: similar pairs are marked with a ' + ' label and dissimilar pairs with a ' -'. The objective is to partition the nodes to minimize disagreements-i.e., cases where the partitioning contradicts the edge labels. Due to its flexibility in not requiring a predefined number of clusters, CC has been widely utilized in various areas such as detecting communities in networks [17], inferring labels from user interactions [1, 13] and resolving ambiguous entities [28].

Dahoon Lee

Department of Mathematical Sciences Seoul National University Seoul, 08826, South Korea dahoon46@snu.ac.kr

However, classic CC models only binary relationships, which is insufficient for many practical applications. For example, in a social network, edges may represent diverse relationship types such as 'colleague,' 'classmate,' or 'family.' To address this limitation, Bonchi et al. [10] introduced the Chromatic Correlation Clustering (CCC) problem, which generalizes CC to multi-class categorical settings. In CCC, the input is an edge-colored graph where each color represents a different relationship type. The goal is to cluster the nodes and assign a single color to each cluster such that the number of disagreements -edges whose color does not match the cluster's assigned color, or edges that should be separated-is minimized. CCC has wide applications in link classification, entity resolution, and clustering in bioinformatics [10, 3, 30].

In parallel, another important generalization of CC is the weighted correlation clustering problem , where edges are associated with weights reflecting the reliability or cost of violating a given label. When weights are unrestricted, obtaining a constant-factor approximation is known to be hard (under the Unique Games Conjecture) [29]. However, when edge weights form a pseudometric -i.e., they satisfy the triangle inequality-constant-factor approximations become feasible. This weighted setting more faithfully models scenarios where not all edges are equally trustworthy.

1.1 Related Works

The Correlation Clustering problem has been widely studied since its introduction [9], and it is known to be APX-hard, leading to efforts to develop approximation algorithms. Early work by Bansal, Blum, and Chawla introduced a constant-factor approximation algorithm [5]. Charikar et al. [15] improved this to a 4-approximation using linear programming. Ailon, Charikar, and Newman then introduced the Pivot algorithm [2], which achieved a 3-approximation in linear time. Chawla et al. [16] further improved this to 2.06 using more refined LP-rounding techniques. More recently, researchers have surpassed the 2-approximation barrier. Cohen-Addad, Lee, and Newman [23] used the SheraliAdams hierarchy to develop a (1 . 994 + ε ) -approximation, while Cohen-Addad et al. [22] proposed preclustering , which improved the approximation to (1 . 73 + ε ) . The most recent breakthrough by Cao et al. [12] introduced the cluster LP , which unifies all known LP relaxations for CC. They show that this can be approximated efficiently using preclustering, achieving a (1 . 437 + ε ) -approximation, the best known guarantee for CC so far. In a more recent work [12], they introduced a new approach to find a feasible solution for the cluster LP in sublinear time.

Chromatic Correlation Clustering is an extension of the classical Correlation Clustering problem, where edge colors represent different types of relationships. Bonchi et al. [10] introduced CCC with a heuristic lacking guarantees. Anava et al. [3] gave a 4-approximation via LP rounding, plus two practical methods: Reduce and Cluster (RC, ratio 11) and Deep Cluster (DC). Klodt et al. [30] showed that Pivot [2] yields a 3-approximation and that RC achieves a 5-approximation. More recently, Xiu et al. [32] developed a 2.5-approximation algorithm for CCC based on a linear programming approach, improving upon the previous best-known ratio. They also introduced a greedy heuristic that achieves strong empirical results.

In modern data analysis, correlation clustering must often be performed under computational constraints such as limited memory or streaming access to data. Consequently, substantial research has focused on crafting clustering algorithms specifically tailored for dynamic, streaming, online, and distributed settings [31, 25, 27, 18, 26, 19, 4, 21, 7, 8, 6, 20, 11].

1.2 Our Results

Our Contributions. In this work, we present improved approximation algorithms for both the CCC and pseudometric-weighted CC problems.

For the pseudometric-weighted correlation clustering problem, we develop a refined LPbased pivoting algorithm that achieves a tight 10 3 -approximation . We further prove that this approximation factor is optimal within the standard LP relaxation framework combined advanced rounding functions.

For the Chromatic Correlation Clustering problem, we enhance the LP-based method through a new analysis that yields a 2 . 15 -approximation , improving upon the previous best bound of 2 . 5 by Xiu et al. [32]. We also establish a lower bound of 2 . 11 within the same analytical framework, underscoring the near-optimality of our approach.

Both results are obtained by extending and unifying the triple-based analysis of LP-rounding schemes. Our work improves the theoretical guarantees for two natural and practically motivated generalizations of correlation clustering and contributes new insights into their structural and algorithmic properties.

Technical Overview. Our algorithms for both Chromatic Correlation Clustering (CCC) and pseudometric-weighted Correlation Clustering (CC) build on linear programming (LP) relaxations and a unified triple-based rounding framework [16] . Below, we outline the key technical insights:

Pseudometric-Weighted CC: The upper bound for the approximation factor 10 / 3 is derived using the LP-based Pivot algorithm and a more careful rounding function. For the lower bound, By assuming the existence of an α -approximation and analyzing carefully constructed hard instances, the technique derives necessary conditions that any rounding function must satisfy. These conditions expose inherent conflicts, demonstrating that α cannot be arbitrarily small. In particular, the analysis establishes that α must be at least 10 3 . The core idea is to identify instance configurations that induce contradiction between the properties required of the rounding functions, ultimately leading to this lower bound on α .

Chromatic Correlation Clustering (CCC): Building on the LP formulation introduced by Xiu et al. [32], which jointly encodes fractional cluster membership and color assignments. The decoupling of color assignment from cluster formation, allowing us to preserve color structure without entangling it with clustering decisions. Using a triple-based analysis, we introduce tailored rounding functions-particularly for neutral edges-to better align the rounding behavior with the LP's structure and avoid overcounting. This careful handling of intra-color, conflicting, and neutral edges reduces the approximation factor from 2.5 to 2.15. Our lower bound analysis builds on the general triple-based framework, augmented with structural insights specific to the LP-CCC algorithm and its associated LP solution. We carefully define the cost and LP contribution of each edge type-particularly neutral edges-and construct adversarial instances that expose limitations of any rounding strategy.

Paper Organization. The remainder of the paper is structured as follows: Section 2 introduces the problem formulations and LP relaxations for both pseudometric-weighted and chromatic correlation clustering. Section 3 presents our approximation algorithms and outlines their design. Section 4 defines the rounding functions used in the LP-based algorithms. Section 5 provides a detailed triple-based analysis of the approximation guarantees. We conclude with a summary and discussion in Section 6 .

2 Preliminaries

The correlation clustering (CC) problem takes as input a signed undirected graph G = ( V, E = E + ⊎ E -) , where each edge e = uv ∈ E is assigned a sign ' + ' or ' -', described by e ∈ E + or e ∈ E -. The objective is to find a partition of the nodes such that the number of disagreements -i.e., negative edges within the same cluster and positive edges between different clusters-is minimized. In other words, the cost of the clustering C is as follows:

where x uv = 0 indicates that there exists C ∈ C such that u, v ∈ C , and x uv = 1 otherwise.

CC has a standard LP relaxation leveraging the viewpoint on x as a discrete metric between partitions. Since the x above satisfies the triangle inequality, we can relax the range of x from { 0 , 1 } to [0 , 1] , resulting in the following LP:

The integrality gap of CC-LP on a complete graph is known to be 2 [15], which indicates that the standard LP-based algorithm cannot obtain a better approximation factor below 2 .

2.1 Pseudometric-weighted Correlation Clustering

The weighted Correlation Clustering problem is a generalization of the classical CC problem in which each edge is associated with a nonnegative violation cost. Specifically, for each edge uv in a complete graph, a weight w uv ≥ 0 is provided, and violating the edge's label (either ' + ' or ' -') incurs a penalty of w uv . This differs from the standard setting, where all violations incur a uniform cost of 1 . The weighted variant allows us to encode edge-wise reliability: when w uv is large, it is more reasonable to follow the given label between u and v .

However, assuming the Unique Games Conjecture , no O (1) -approximation algorithm exists for the general weighted case [24]. An exception occurs when the weight function satisfies the triangle inequality, i.e., the weights form a pseudometric . In this pseudometric-weighted setting, a constantfactor approximation is known [14]. Following the analysis of Charikar and Gao with L = 2 yields an approximation factor of B HR + 1 1 3 ≤ 4 1 3 +2( L -1) + 1 1 3 = 17 , since the second type of charge occurs at most L -1 = 1 time in the charging scheme. The following is a natural LP relaxation of the weighted CC problem, extending (CC-LP):

Here, the variable x can be viewed as defining a pseudometric over the vertex set, representing the distance between clusters. Since CC on bipartite graphs has an integrality gap of 3 [16], and is a special case of pseudometric-weighted CC, the LP relaxation (wCC-LP) for pseudometric-weighted CC also has an integrality gap of at least 3.

2.2 Chromatic Correlation Clustering Problem

The Chromatic Correlation Clustering problem is a variant of the classical CC problem in which each cluster is additionally assigned a color [10]. The input includes a complete graph ( V, ( V 2 ) ) and a set of L possible colors, as well as a special color γ that denotes that two vertices should not be placed in the same cluster-analogous to a negative (' -') edge in the classical CC setting. When L = 1 (i.e., a single cluster color), CCC reduces to the standard CC problem with a complete instance.

The following is a linear programming (LP) relaxation of the CCC problem [10]:

̸

Here, the variables x c u and x c uv are soft assignments:

1 -x c u ∈ [0 , 1] represents the fractional assignment of vertex u to a cluster of color c .

1 -x c uv ∈ [0 , 1] indicates the fractional agreement between vertices u and v under color c .

These variables measure the likelihood of vertices or edges being assigned to a color, with { 1 -x c u } c ∈ L forming a probability distribution over the colors assigned to vertex u , subject to constraints (7) and (8).

There is also a geometric interpretation of these variables. Consider L discrete pseudometric spaces ( V c , d c ) where V c = { u c : u ∈ V } , and vertex u is connected to u c with a link of length x c u . Then,

x c uv represents the bottleneck distance between u and v , conditioned on traversing the auxiliary connections u → u c and v → v c . This view generalizes the classical CC setting, where the cluster-wise discrete metric can be regarded as a special case of bottleneck distances.

Since CCC generalizes the standard CC problem, the integrality gap of (CCC-LP) is at least as large as that of (CC-LP), which is 2.

3 Approximation Algorithm

Building on the LP formulations introduced in the previous sections, we now present approximation algorithms for both pseudometric-weighted CC and CCC settings. LP-PIVOT (Algorithm 1) extends the classical LP-based pivoting method. The set of edges is divided into 3 subsets: E + and E -indicate a set of '+ ′ and ' -′ edges, respectively, while E ◦ indicates a set of edges that always incur a cost regardless of the output. The last subset is involved in the CCC case, as some of the edges might already be misclassified before the execution. f + , f -, and f ◦ are rounding functions, which are explained in Section 4. The time complexity of the algorithm is O ( | V | 2 ) .

Algorithm 1 LP-PIVOT

The algorithm for the pseudometric-weighted CC problem is LP-PIVOT (( V, E + ⊎ E -⊎ ∅ ) , { x ∗ uv } uv ∈ ( V 2 ) ) along with selected rounding functions given by equation (11). The time complexity of the algorithm is dominated by solving the LP, which is polynomial in | V | .

The algorithm for the CCC problem is LP-CCC ( G,ϕ,x ) (Algorithm 2) along with the differently selected rounding functions given by equations (12) and (13), which first partitions the vertices according to their LP-derived color distributions, followed by applying the LP-PIVOT algorithm, with edge types partitioned by color. The final clustering is obtained by combining the | L | number of outputs from the LP-PIVOT algorithm. The color pre-classification step requires O ( | V || L | ) time and the following LP-PIVOT step requires at most O ( | V | 2 ) time in total, which are both dominated by the time complexity of solving (CCC-LP), which is polynomial in | V | and | L | .

4 Rounding Functions

The effectiveness of the LP-PIVOT and LP-CCC algorithms critically depends on the choice of rounding functions used in the clustering process. Rounding functions f + , f -, f ◦ : [0 , 1] → [0 , 1] convert the LP value x uv to the non-selection probability p uv [16]. The sign of the edge uv -either ' + ', ' -', or ' ◦ '-determines which rounding function is applied. The sign ' ◦ ' indicates that the edge does not belong to E . The following natural conditions are imposed on any rounding function f :

Algorithm 2 LP-CCC

Condition 9 is not only intuitive but also necessary in certain cases, such as ensuring f + (0) = 0 and f -(1) = 1 . Other constraints are not required in the proofs of Theorems 1 and 3, which thus provide lower bounds on the approximation factors for 'general' rounding functions.

Lemma 1. The LP-PIVOT algorithm achieves a constant-factor approximation in expectation only if f + (0) = 0 and f -(1) = 1 .

Proof. We prove it by contradiction on some graph instances.

Case 1 f + (0) = 0 . Consider G = ( V, E = ( V 2 ) = E ⊎ ∅ ⊎ ∅ ) . The optimal clustering is C ∗ = { V } , satisfying obj ( C ∗ ) = 0 , and the optimal LP solution is x ∗ ≡ 0 .

̸

Suppose f + (0) > 0 . Then Pr[ LP-PIVOT ( G, 0) = C ∗ ] > 0 . Since obj ( C ) > 0 if and only if C ̸ = C ∗ , this leads to a contradiction with the assumption of the expected constant factor approximation.

Different variants of the CC problem may use different rounding functions. In this paper, we provide rounding functions for both the pseudometric-weighted CC and CCC problems.

4.1 Pseudometric-weighted Correlation Clustering

We propose the following rounding functions that yield a tight approximation factor:

With these functions, the algorithm achieves an expected approximation factor of 10 / 3 . Moreover, no other rounding function can improve this factor, as shown in Section 5.1.

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

x

y

Prevented Regions vs Rounding function

4.2 Chromatic Correlation Clustering

We further consider the rounding functions f + , f -from Chawla et al. [16], which yield a 2 . 06 -approximation for classical CC, and introduce a new function f ◦ to handle ◦ -edges for CCC:

This function was designed not to intersect with analytic bounds that violate an approximation factor of α = 2 . 15 , as illustrated in Figure 1. From Figure 1, x refers to the LP value corresponding to the pivot edge (i.e., one of the endpoints is a pivot vertex) whose color differs from the color under execution of the pivot-based algorithm, while y refers to the corresponding probability of not containing the edge in a single cluster. The plot shows the choice of our f ◦ and the region of ( x, y ) such that: The triple-based analysis in Section 5 cannot guarantee an α = 2 . 15 -approximation if the value x is assigned to the probability of y by the rounding function f ◦ .

4.3 Comparison with Prior Work

Pseudometric-weighted CC: The LP-UMVD-PIVOT algorithm, recently proposed by Charikar and Gao [14] for the Ultrametric Violation Distance (UMVD) problem, follows a pivoting-based rounding strategy applied to an LP relaxation. When the number of distinct pairwise distances between elements, denoted by L , is equal to 2, their algorithm can be viewed as a special case of our LP-PIVOT framework. In this setting, the two distances d 1 and d 2 correspond to the ' -' and ' + ' labels, respectively, used in our rounding procedure. The rounding functions used are:

Here, α and β are fixed algorithmic parameters. For the pseudometric-weighted CC problem, they are set to α = 1 3 and β = 0 , and f ◦ is unused.

With triple-based analysis, this choice yields an approximation factor of 6 , as shown in the subsection ?? .

Chromatic CC: The LP-based pivoting algorithm by Xiu et al. [32] uses LP values directly as probabilities, corresponding to the following rounding functions:

This setting is known to achieve an approximation factor of 2 . 5 .

5 Triple-based Analysis

To complete the analysis of the algorithm, it suffices to show that for every triple of vertices u, v, w ∈ V , the expected cost incurred by the algorithm, denoted ALG ( uvw ) , is at most a factor α times the corresponding LP cost LP ( uvw ) . That is,

If the inequality holds for every triple, then the total expected cost of the algorithm is at most α · LP . To show this, the analysis expresses the expected algorithmic cost and LP cost as averages over all possible pivot choices and vertex triples. Specifically, it defines:

e.cost w ( u, v ) : the expected cost of violating constraint ( u, v ) , conditioned on pivot w ,

e.lp w ( u, v ) : the expected LP charge of edge ( u, v ) , conditioned on pivot w .

Here, LP charge is an event that either of the endpoints of the edge is gathered with the pivot vertex, multiplied by the LP value of the edge. Since charging occurs exactly once for each edge, accumulating every charge results in exactly the LP cost.

5.1 Pseudometric-weighted Correlation Clustering

In the CC setting, we use the function C , as defined in [16], to measure the gap:

where and

In the weighted CC setting, edge weights further influence the value of C :

with the definition for e.cost and e.lp remains the same; the classical CC corresponds to ( w uv , w vw , w wu ) = (1 , 1 , 1) .

Under the pseudometric constraint on weights w , we can reduce the number of cases to consider in the analysis.

Lemma 2. If α · LP ( uvw ) -ALG ( uvw ) ≥ 0 holds for weight configurations ( w uv , w vw , w wu ) ∈ { (1 , 1 , 0) , (1 , 0 , 1) , (0 , 1 , 1) } , then the inequality also holds for any configuration ( w uv , w vw , w wu ) satisfying the triangle inequality.

Proof. Let all x uv , x vw , x wu , p uv , p vw , p wu be fixed. ALG ( uvw ) and LP ( uvw ) can be written as

and

Therefore, the function αLP ( uvw ) -ALG ( uvw ) is linear w.r.p. ( w uv , w vw , w wu ) .

Since the set of ( w uv , w vw , w wu ) that satisfies the triangle inequality forms a convex cone generated by (1 , 1 , 0) , (1 , 0 , 1) , (0 , 1 , 1) , the function value is nonnegative for all such ( w uv , w vw , w wu ) if and only if the value is nonnegative for ( w uv , w vw , w wu ) ∈ { (1 , 1 , 0) , (1 , 0 , 1) , (0 , 1 , 1) } .

This lemma implies that the algorithm achieves an approximation factor of α if all of the following inequalities are satisfied for every possible configuration on the triangle uvw :

We obtain a lower bound on the approximation factor of LP-PIVOT by verifying the feasibility of rounding functions that satisfy the above inequalities. To this end, we analyze several configurations of LP values and edge signs on triangle uvw .

In Theorems 1 and 3, the notation ' ( a, b, c ) with ( s 1 , s 2 , s 3 ) ' denotes ( x uv , x vw , x wu ) = ( a, b, c ) , where each edge sign is given by s 1 , s 2 , and s 3 , respectively.

Theorem 1. The lower bound on the approximation factor of LP-PIVOT in pseudometric-weighted correlation clustering is 10 / 3 . The proof is deferred to the Appendix.

Conversely, there exist rounding functions f + , f -making the approximation factor of LP-PIVOT by 10 / 3 , providing that the lower bound above is tight.

Theorem 2. The LP-PIVOT algorithm with the rounding function defined in equation 11 yields a 10 / 3 -approximation algorithm for pseudometric-weighted CC. The proof is deferred to the Appendix.

5.2 Chromatic Correlation Clustering

We analyze the performance of the LP-CCC algorithm. This algorithm begins by assigning each vertex to its majority color based on the LP solution, followed by a pivot-based clustering routine.

Due to the strict majority condition, any edge not included in ⊎ E c must have an LP value of at least 1 / 2 . Thus, the cost incurred by such edges is at most twice their LP contribution [32].

Within each color class S c , corresponding to color c , we follow an analysis similar to that of Chawla et al. [16]: edges of color c are treated as positive edges ( E + ), edges of the adversarial color γ as negative edges ( E -), and all other edges as neutral ( E ◦ ).

For positive and negative edges, the definitions of e.cost and e.lp remain consistent with those in [16]. The other three cases, particularly those involving neutral edges, require more careful treatment.

Consider a negative edge uv ∈ E -: the LP value is

For a neutral edge uv ∈ E ◦ , the expected cost arises from the event that u and v are not separated by w , i.e., at least one of them shares a cluster with w . The expected cost is thus given by the probability that u and v are not simultaneously separated from w .

̸

The LP contribution in this case is the product of this probability with x ϕ ( uv ) uv , where ϕ ( uv ) = c is the color of edge uv in the input. While x ϕ ( uv ) uv is not tied to color c , we can still bound it below using

x c uv , x c vw , x c wu due to LP constraints [32]:

Summarizing the results, we express the expected cost and lower bound on the LP value for a fixed pivot w as follows:

These formulations are central to the analysis. Since α · LP -ALG is always at least the expression obtained from the LP lower bound, we can prove that this bound is nonnegative.

As in Section 5.1, the algorithm achieves an α -approximation if the following inequality holds for all triangles uvw :

This inequality leads to the following result on the approximation guarantee for LP-CCC:

Theorem 3. The approximation factor of LP-CCC for CCC is bigger than 2 . 11 . The proof is deferred to the Appendix.

Analogous to the classical CC setting-where the lower bound and the approximation ratio of LPPIVOT differ by less than 0 . 04 [16]-augmenting the LP rounding with a suitable f ◦ yields the following:

Theorem 4. LP-CCC , using the rounding functions in (12) and (13) , achieves a 2 . 15 -approximation for Chromatic Correlation Clustering. The proof is deferred to the Appendix.

6 Conclusion

In this work, we studied two important variants of correlation clustering: pseudometric-weighted correlation clustering and chromatic correlation clustering . For both problems, we developed and analyzed specialized rounding functions that are essential for achieving improved approximation guarantees via the LP-PIVOT algorithm.

For the pseudometric-weighted setting, we proposed a piecewise-linear rounding function tailored for the setting that achieves a 10 / 3 -approximation, and showed that no alternative function within our analytical framework can yield a better factor. For the chromatic correlation clustering variant, we designed a distinct rounding function that respects the constraints imposed by color restrictions and achieves an approximation factor of 2 . 15 . The function is constructed using a piecewise-linear form and leverages a careful analysis of triple costs.

Overall, our work highlights the importance of designing principled and variant-specific rounding strategies to extend LP-based techniques to structured clustering problems, yielding strong theoretical guarantees.

Acknowledgment

This work by CF and DL was partially supported by the New Faculty Startup Fund at SNU.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studied polynomial-time algorithms for the pseudo-metric correlation clustering and the chromatic correlation clustering problems. The problems are natural generalizations of the classical correlation clustering problems (with (+) and (-) edges and possibly weights). \n\n- In the pseudo-metric correlation clustering problem, our goal is to put vertices of a given graph $G=(V,E)$ to clusters such that the total number of disagreements, defined as the total *weights* of (+) edges crossing clusters and (-) edges inside the same clusters, are minimized. In general, it is impossible to obtain poly-time algorithms for correlation clustering on weighted graphs under plausible hardness assumptions. The paper instead considered the case when the weights satisfy triangle inequalities and obtained a $10/3$-approximation.\n- In the chromatic CC problem, we are given an associated color with each edge (or a label of (-)), and our goal is to produce clusters such that the total number of color disagreements and (-) edges inside the same clusters is minimized. For this problem, the paper obtained a $2.15$ approximation, which improved upon the previous 2,5 approximation given by XHTCH [NeurIPS’22].\n\nFor both problems, the paper gave lower bounds for the specific algorithms (using the LP integrity gap). \n\nThe core techniques for both problems are linear programming (known in the literature) and new rounding algorithms (which are the main novelty of the paper). With fractional random variables, we can obtain the exact answer in polynomial time for the corresponding linear programs for both problems. To produce an integral solution, one of the popular techniques is the pivot-based rounding: we pick a vertex $w$ uniformly at random, and for each neighboring vertex $u$, we add $u$ to the cluster of $w$ based on the variable $x_{uw}$ and the edge label of $(u,w)$. The main technical contributions of the paper are new rounding functions $f^+$, $f^-$, and $f^\\text{neutral}$ for different edge types. The paper adopted a region-based analysis to show the upper bound for the approximation algorithm, which appears to follow the same idea as in CMSY [STOC’15].\n\nSTRENGTHS AND WEAKNESSES:\nI read the first 9 pages of the paper to some fair amount of detail, and I briefly looked into the appendix. My general opinion of the paper is positive: I agree that the pseudometrics CC and CCC are important generalizations for the correlation clustering problem with a wide range of potential applications. The algorithms, while using ideas similar to the ones in the literature, also contain non-trivial technical aspects. I unfortunately did not get a time to check the correctness in detail; a spot check found no issues on this front.\n\nSome weaknesses I want to flag:\n- The paper assumes access to the LP solution, which might not be very efficient in practice. Maybe the author could comment on how fast could one solve the LPs listed in this paper (a recent STOC’25 paper showed “correlation clustering LP” (not the one listed in this paper) could be solved in $O(n \\text{polylog}(n))$ time – can you get similar guarantees?)\n- The concerns about efficiency could also be alleviated by some experiments. Unfortunately, the paper does not contain experiments.\n- I also think the paper could do a much better job on the *accessibility* front. I’m fairly familiar with the topic, so I don’t find many issues reading the results. However, when reading the analysis, I was confused why the charges on the triangles should be $\\sum_{(u,v) \\in E^+}(1-p_{uw}p_{vw})x_{uv}+sum_{(u,v) \\in E^-}(1-p_{uw}p_{vw})(1-x_{uv})$. In fact, the definition of “charge”  was never introduced in this paper, and I do not think the general audience will be able to follow (I had to read the section in CMSY [STOC’15] to remind myself why it’s the case). \n\nOverall, I’m supportive of the paper and will be happy to see it get accepted.\n\nQUESTIONS:\nSome of the questions were asked when I was flagging the weaknesses. Some additional questions and comments (comments are mostly about presentations):\n- Since your lower bound proofs (for both the pseudometric CC and CCC) are based on integrality gaps, I assume one could not hope for better rounding algorithms based on these LPs, am I correct?\n- If yes, then do you believe the more recent correlation clustering LPs, e.g., the one in “Understanding the Cluster LP for Correlation Clustering” [STOC’24] will be helpful for further improvements?\n- I do not understand Figure 1. Here, what are the x and y axes? What does the figure want to say? I hope it could give some insights for the choice of the intricate constants.\n- Notations $E^\\circ$ and $f^\\circ$ were used before being defined.\n- Missing parentheses for equation (13) on the caption of Figure 1.\n- In the appendix, you used tables to show that by the choice of $\\alpha$, we have $\\alpha\\cdot LP - ALG \\geq 0$. The captions of these tables are “caption” – change this.\n\nLIMITATIONS:\nThe paper is theoretical in nature, and I do not see any immediate negative societal impacts.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-06-27"}

comments: SUMMARY:
This paper studied two variants of correlation clustering, i.e., pseudometric-weighted correlation clustering and chromatic correlation clustering, and proposed specialized rounding functions for these two problems to achieve improved theoretical approximation guarantees.

STRENGTHS AND WEAKNESSES:
S1. The contributions of this work is clearly stated and justified. The comparison with previous work is provided.

S2. The contributions of this work are theoretically solid. Improved approximation guarantees are achieved for Pseudometric-Weighted CC and CCC problems.

W1. The paper is heavy in mathematical notations. There is a lack of intuitive understanding of the rationale of the proposed solution.

W2. No empirical results are provided to validate the theoretical outcome.

W3. Time complexity analysis is not given.

QUESTIONS:
1. What is the time complexity of the proposed algorithm?
2. Can we extend the proposed algorithms to solve other variants of correlation clustering problems and achieve improved approximation guarantees? If no, what are the difficulties?

LIMITATIONS:
The authors did not explicitly pinpoint the limitations of the work.
TITLE: Official Review
CLARITY: 3
IMPACT: 2
ORIGINALITY: 2
DATE: 2025-07-01

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe manuscript addresses two variants of the correlation clustering problem. The first variant is clustering correlation with pseudo metric weights. The second variant is chromatic clustering correlation. In both cases the manuscript provides approximation algorithms based on LP relaxations. Both algorithms provide optimal, or near optimal, results (within the LP framework). On the one hand, the novelty and originality of the paper are somewhat limited, in the sense that the methods and arguments are fairly standard, and the authors are able to obtain improved results via a detailed case analysis and very careful selection of certain rounding functions. On the other hand, the theoretical results improve the state of the art, and this is a significant achievement.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: \nS1. The problem of correlation clustering and its variants have been studied extensively in the literature of theoretical computer science. The current manuscript makes a good improvement of the state of the art results, for the approximation ratio of the problems under study.\nWeaknesses:\nW1. The methods and ideas are not so novel. The improvement has been possible via a detailed case analysis and careful selection of rounding functions. The resulting algorithm may give improved quality guarantee but it is not so novel and not so elegant.\nW2. The methods rely on LP relaxation, which is not so practical. Overall, this is a theory paper and there are no experiments.\nW3. In terms of presentation, it is not always so clear when the authors refer to the problem where all the edges are present (clique) and when the edges may be missing (no + or - signs, or no color labels).\n\nQUESTIONS:\nI would be interested to see answers to the weak points, W1--W3, mentioned above.\n\nLIMITATIONS:\nI did not find much discussion on limitations. My overall impression is that the authors do not make a significant effort to offer a critical assessment of their work along different dimensions.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-01"}