Improve the approximation guarantee for Subset Feedback Vertex Set

Improve the approximation guarantee for Subset Feedback Vertex Set beyond the known factor-8 upper bound, toward determining its precise approximability.

Background

Subset Feedback Vertex Set generalizes Feedback Vertex Set by requiring the deletion of a minimum-cost set of vertices that intersects every cycle containing a terminal. Its approximability is unresolved: the known lower bound is 2, inherited from Feedback Vertex Set, while the best-known upper bound is 8. The paper specifically identifies improving the upper bound supplied by the existing compact LP relaxation as an open problem.

References

Chekuri and Madan described a compact LP relaxation for SFVS with an integrality gap of at most $13$, and it remains open to improve this upper bound.

— 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 “Motivations for the Extreme Point Conjecture” and Section 8, Conclusion