Efficient separation for the Strong-Density Polyhedron

Determine an efficient separation oracle for the Strong-Density polyhedron P_SD(G) for Feedback Vertex Set.

Background

The Strong-Density polyhedron P_SD(G) has exponentially many constraints and is used to derive an iterative-rounding 2-approximation for Feedback Vertex Set. Although the paper develops a round-or-cut procedure that yields a polynomial-time algorithm through the Ellipsoid method, it does not provide a general efficient separation oracle for P_SD(G). The authors explicitly identify this missing algorithmic capability, contrasting it with the Edge-Strong-Density polyhedron, for which a polynomial-time separation oracle is established.

References

Note that we do not yet know an efficient separation oracle for $P_{SD}(G)$, which perhaps suggests that the stronger relaxation $P_{Edge-SD}(G)$ is more natural in a certain sense.

— An iterative rounding $2$-approximation for Feedback Vertex Set via AI-assisted proof of an extreme point property  (2609.04414 - Chandrasekaran et al., 3 Sep 2026) in Section 1, paragraph “Solving the LP Relaxations and Iterative Rounding Algorithms”