One Color Preprocessing Improves DSATUR

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.

Gray-box optimization is an approach for making some problem-specific information available to the algorithm while still relying on fitness information as the main guide to an optimum. This approach was shown to be beneficial in various combinatorial optimization tasks and neatly captures the continuum between fully black-box algorithms and tailored algorithms. In this work, we discuss different flavors of gray-box algorithms. We show that RLS can find a proper $2$-coloring in a bipartite graph starting from a random $2$-coloring, in an expected time of $\mathcal{O}(n \log n)$. In contrast, when starting from a proper $n$-coloring, the (1+1) EA cannot find such a coloring except when offered additional guiding on plateaus of the search space. Finally, we show the run time for this setting can be much improved by using gray-box operators.

The stable coloring of the Weisfeiler-Leman (1-WL) test is a cornerstone of Graph Neural Networks because it provides an upper bound to the expressive power of message-passing architectures. Unfortunately, computing it presents two fundamental bottlenecks. First, classic algorithms are inherently sequential and cannot exploit modern massively parallel hardware. Second, these are \emph{global} algorithms, i.e., they require availability in memory of the full graph, severely limiting applicability to real-world instances. We leverage a linear-algebraic interpretation of 1-WL stable coloring and introduce two key contributions: (i)~a randomized refinement algorithm with tight probabilistic guarantees and (ii)~a correctness-preserving batching scheme that decomposes the graph into independently processable subgraphs while provably returning a stable coloring of the original graph. This approach maps directly to GPU-efficient primitives. In numerical experiments, our CUDA implementation delivers speedups up to two orders of magnitude over classical CPU-based partition refinement and, for the first time, successfully computes stable colorings on web-scale graphs with over 30 billion edges, where CPU baselines time out or fail.