Universal 2.030-Rounding for Multilayer Correlation Clustering with Probability Weights

Multilayer correlation clustering seeks one partition for conflicting preferences across layers. For probability weights, where joining and separating each pair have costs summing to one, we establish universal $2.030$-rounding. Specifically, any explicit feasible rational cluster-LP cover can be rounded deterministically to one partition costing at most $2.030$ times the cover's fractional cost for all probability-weight vectors simultaneously, including those not supplied to the algorithm. We achieve this through a single auxiliary objective and potential, conditional variance bounds, and computer-assisted verification using an exact rational certificate. Consequently, combining this rounding with scalar fractional optimization and averaging yields a randomized $(2.030+ϵ)$-approximation for each fixed ${\ell }_p$ objective, $p\in [1,\infty ]$, with a factor independent of the number of layers. Together with a two-vertex lower bound, the rounding theorem places the worst-layer cluster-relaxation gap in $[2,2.030]$.