Positive-correlation rounding for uncorrelated in-links while preserving marginals

Develop a sampling or rounding scheme for the Structured LP/Strong LP-based approach that positively correlates the selections on the two leading edges of each uncorrelated in-link while maintaining the prescribed marginals on each edge, thereby reducing the probability of selecting the in-link on only one side.

Background

In the paper’s rounding approach, uncorrelated in-links can be sampled independently on their two leading edges, causing a factor-2 cost contribution from such links. Achieving positive correlation between the two sides of an in-link—while preserving edge-wise marginals—would reduce this cost by increasing the likelihood that either both or neither side selects the in-link.

The authors outline the intuitive goal of correlating the distributions over the two edge-level selections but report that they do not know how to accomplish this, noting that devising such a strategy would be significantly beneficial for further improving the approximation.

References

Intuitively, to do so, we would like to positively correlate the distributions over $L_e$ and $L_f$ for every in-link $\ell$ with leading edges $e, f$ so that if $\ell$ is taken on one side, it is more likely to be taken on the other as well (while maintaining the marginals on each side). Unfortunately, we do not know how to do this.

— A Strong Linear Programming Relaxation for Weighted Tree Augmentation  (2603.29582 - Cohen-Addad et al., 31 Mar 2026) in Section 4 (Cleanup Phase)