Chromatic Cluster LP Relaxation

• The paper introduces the chromatic cluster LP relaxation that indexes variables by vertex subsets and colors to directly model fractional colored clusters in CCC.
• It utilizes three-node exclusivity constraints and advanced rounding schemes to counteract fractional splitting across colors, achieving approximation ratios as low as 1.64.
• The framework offers a polynomial-time (1+ε)-approximate LP solver using Sherali–Adams lifts, enabling efficient rounding and improved performance over standard pairwise LPs.

Searching arXiv for papers on chromatic cluster LP relaxation and related CCC literature. Chromatic cluster LP relaxation is an exponential-size linear programming framework for Chromatic Correlation Clustering (CCC) in which variables are indexed by subsets $S\subseteq V$ and colors $c\in L$, so the relaxation reasons directly about fractional colored clusters rather than only pairwise separations. In 2025, this framework was used to obtain a randomized $1.64$-approximation for CCC and, in a separate line, a randomized $(2+\varepsilon)$-approximation together with a polynomial-time $(1+\varepsilon)$-approximate LP solver (Lee et al., 21 Jul 2025, Abbasi et al., 15 Oct 2025). Its role is clearest when contrasted with the standard CCC LP: that pairwise relaxation admits a $2.15$-approximation, has a strict lower bound of $2.11$, and incurs an additional chromatic penalty under color-independent rounding (Lee et al., 28 May 2025, Shihab et al., 17 Apr 2026).

1. Problem setting

CCC generalizes ordinary correlation clustering by replacing binary $+/-$ edge semantics with a color set $L$ of positive labels and a distinguished negative label $\gamma$. A typical formulation takes a complete graph whose edge labels satisfy $c\in L$0. A feasible integral solution is a partition $c\in L$1 together with a cluster-color assignment $c\in L$2. The disagreement objective is

$c\in L$3

This formulation captures the requirement that a cluster must carry a single chromatic label while positive edges prefer co-clustering only in the matching color (Lee et al., 21 Jul 2025).

The chromatic constraint is the source of the additional difficulty. In ordinary correlation clustering, a $c\in L$4 edge only asks to be kept inside some cluster, whereas in CCC a colored positive edge asks to be kept inside a cluster of the correct color. As a result, edges of other colors become structurally relevant even when two vertices are co-clustered. This distinction is central both to the failure modes of pairwise LPs and to the motivation for subset-based relaxations (Shihab et al., 17 Apr 2026).

2. Subset-based LP formulations

One 2025 formulation introduces a variable $c\in L$5 for every nonempty $c\in L$6 and every color $c\in L$7, intended to be the fractional number of times $c\in L$8 appears as a cluster of color $c\in L$9. It then defines

$1.64$0

and minimizes

$1.64$1

The constraints are: $1.64$2

$1.64$3

$1.64$4

together with $1.64$5 (Lee et al., 21 Jul 2025).

A second 2025 presentation, due to Abbasi et al., uses the same subset-and-color indexing but writes the LP directly in $1.64$6-space: $1.64$7 subject to

$1.64$8

Here $1.64$9 is the set of positive edges crossing $(2+\varepsilon)$0, and $(2+\varepsilon)$1 is the set of edges inside $(2+\varepsilon)$2 that are either negative or positive of a color different from $(2+\varepsilon)$3. This formulation introduces derived variables

$(2+\varepsilon)$4

so the notational polarity of the $(2+\varepsilon)$5-variables depends on the paper: one presentation defines them as fractional coverage, the other as fractional separation (Abbasi et al., 15 Oct 2025).

When $(2+\varepsilon)$6, the chromatic LP collapses to the usual monochromatic cluster LP of Charikar–Guruswami–Wirth. In that sense, the chromatic formulation is a genuine extension of the cluster-LP paradigm rather than a separate modeling device (Abbasi et al., 15 Oct 2025).

3. Structural strengthening over the standard CCC LP

The standard CCC LP uses only vertex-color variables and pairwise color-plane variables, together with domination, triangle, and chromatic-sum constraints. In the formulation summarized in the literature, it enforces

$(2+\varepsilon)$7

This is the natural chromatic analogue of the classical metric LP, but it remains pairwise and therefore cannot directly encode cluster-level exclusivity (Lee et al., 28 May 2025).

The chromatic cluster LP strengthens this picture in two ways. First, the variables $(2+\varepsilon)$8 represent entire candidate clusters rather than only marginals over edges or vertices. Second, the additional three-node condition

$(2+\varepsilon)$9

forbids simultaneously using two overlapping clusters of the same color on three points. The corresponding $(1+\varepsilon)$0-variable view yields inequalities of the form

$(1+\varepsilon)$1

which were explicitly described as stronger than the standard triangle inequalities (Lee et al., 21 Jul 2025).

This extra structure is aimed at the specific pathology of CCC: fractional splitting across colors. One 2025 account states that the standard LP can be driven down to $(1+\varepsilon)$2 by “splitting” clusters fractionally across colors, while the cluster LP rules out these pathological $(1+\varepsilon)$3 fractional splits by means of subset variables and the “at-most-one-cluster-on-three” constraint (Lee et al., 21 Jul 2025). Another account emphasizes the same point through the internal penalty term $(1+\varepsilon)$4, which charges not only negative edges inside $(1+\varepsilon)$5 but also positive edges of the wrong color (Abbasi et al., 15 Oct 2025).

4. Rounding algorithms

The most direct rounding scheme is a cluster-based peeling procedure. Given a feasible chromatic cluster-LP solution $(1+\varepsilon)$6, it repeatedly samples a pair $(1+\varepsilon)$7 with probability proportional to $(1+\varepsilon)$8, outputs the cluster $(1+\varepsilon)$9 on the currently unassigned vertex set $2.15$0, assigns it color $2.15$1, and removes those vertices. Abbasi et al. prove that this yields a $2.15$2-approximation. Their key bounds are

$2.15$3

and

$2.15$4

which imply $2.15$5 (Abbasi et al., 15 Oct 2025).

A later algorithmic development combines this cluster-based routine with a chromatic pivoting scheme. For each color $2.15$6, it forms the bucket

$2.15$7

then repeatedly chooses a random pivot $2.15$8 and grows a cluster $2.15$9 of color $2.11$0 by including each $2.11$1 with probability $2.11$2, where

$2.11$3

The cluster-based and pivot-based routines are then mixed: with probability $2.11$4 the algorithm runs cluster-based rounding, otherwise pivot-based rounding, and the balancing choice $2.11$5 yields a randomized $2.11$6 approximation in time $2.11$7. The same source states that standard derandomization gives the same guarantee deterministically (Lee et al., 21 Jul 2025).

The pivot component is explicitly presented as an extension of the framework of Cao et al. for ordinary correlation clustering. In the chromatic setting, its novelty is the coexistence of cluster-color selection and cluster formation, mediated through the per-color buckets and the neutral-edge treatment (Lee et al., 21 Jul 2025).

5. Polynomial-time solvability of the exponential LP

Because the chromatic cluster LP has exponentially many $2.11$8 variables, approximation algorithms require a surrogate for exact optimization. Abbasi et al. describe a three-stage scheme that solves the LP to within a factor $2.11$9 in polynomial time and then rounds it to a $+/-$0-approximate integral clustering (Abbasi et al., 15 Oct 2025).

The first stage is preclustering. One computes a constant-factor approximation $+/-$1, marks “bad” vertices that incur many disagreements inside their initial cluster, and breaks large clusters on these vertices. The resulting partition $+/-$2 of $+/-$3 is constructed so that $+/-$4 precluster-adjacencies remain, and there is a near-optimal solution that never merges across non-admissible pairs while respecting the coloring of each precluster (Abbasi et al., 15 Oct 2025).

The second stage is a bounded sub-cluster LP built using Sherali–Adams lifts. The method keeps only clusters $+/-$5 of size at most $+/-$6, enforces that each precluster $+/-$7 is either kept intact or merged only into a sufficiently large cluster, forces all non-admissible edges to be separated, and requires all vertices in a precluster to agree on color. The resulting LP has size $+/-$8 and can be solved in $+/-$9 time (Abbasi et al., 15 Oct 2025).

The third stage samples clusters from the bounded LP solution. Specifically, it draws

$L$0

independent samples, each in time $L$1, and applies a Chernoff-pruning argument so that vertex frequencies and empirical co-occurrence rates match the target LP values up to small error. Defining $L$2 as the fraction of samples that produce $L$3 with color $L$4 yields a feasible solution to the original exponential LP with

$L$5

Rounding then gives total cost at most $L$6 (Abbasi et al., 15 Oct 2025).

6. Integrality gaps, pairwise-LP barriers, and open interpretive issues

The strongest motivation for chromatic cluster LPs comes from the limitations of the standard pairwise relaxation. For the uncoupled CCC LP, the best known LP-rounding algorithm achieves $L$7, and the literature proves a strict lower bound of $L$8 within the same pivot-and-rounding framework (Lee et al., 28 May 2025). A 2026 analysis isolates the obstruction as cross-edge chromatic interference: color-independent rounding pays an irreducible neutral-edge penalty $L$9, and the global integrality-gap decomposition theorem states

$\gamma$0

In particular,

$\gamma$1

The same work shows that adding the global coupling constraint

$\gamma$2

and using the color-coupled $\gamma$3 rounding scheme recovers the optimal $\gamma$4 approximation known for classical correlation clustering (Shihab et al., 17 Apr 2026).

Within the cluster-LP line itself, however, the literature records different gap statements. One 2025 paper states that adding $\gamma$5 variables and the three-node exclusivity constraint causes the integrality gap to collapse to $\gamma$6 (Lee et al., 21 Jul 2025). Another 2025 paper states that the chromatic cluster LP has integrality gap $\gamma$7 even in the chromatic setting, with a two-point graph showing gap $\gamma$8, and therefore that $\gamma$9 is best possible unless stronger relaxations are used (Abbasi et al., 15 Oct 2025). This suggests that “chromatic cluster LP” is being used for closely related but not identically analyzed relaxations, or that the quoted bounds are attached to different formulations and rounding frameworks.

The broader historical context comes from the uncolored problem. In ordinary correlation clustering, the cluster-LP framework introduced by Cao et al. unifies prior relaxations, automatically implies triangle inequalities and PSD-style constraints, admits a simple rounding algorithm with analytical ratio $c\in L$00, and supports a factor-revealing SDP analysis giving $c\in L$01, while also having a $c\in L$02 integrality gap (Cao et al., 2024). A plausible implication is that the chromatic cluster LP should be understood less as a single fixed relaxation and more as a family of subset-based formulations whose strength depends on the exact treatment of color consistency, local overlap constraints, and the rounding interface.