Approximation guarantee for Forest Augmentation beyond the minimum-value regime

Derive an explicit approximation guarantee for the Forest Augmentation Problem when the optimum value of the standard cut LP exceeds the number of zero-cost forest components, by converting the LP-excess budget into a bound that accounts for branching free-forest components and positive-support heavy edges incident with internal path vertices.

Background

The paper establishes a $59/33$ guarantee for Forest Augmentation only in the minimum-value regime, where the cut-LP optimum equals the number of zero-cost forest components. In that regime, equality in the singleton constraints forces every free component to be a path and prevents positive-support heavy edges from being incident with internal path vertices, allowing suppression of the free paths to reduce the instance to Matching Augmentation.

For general Forest Augmentation instances with LP value greater than the number of free components, the paper notes that the free forest may branch and that positive heavy support may meet internal path vertices. The stability inequality shows that these deviations are bounded in aggregate by the LP excess, but the paper does not convert that budget into an approximation ratio.

References

Equations~eq:fap-excess and eq:fap-stability show that both effects are globally paid by the LP excess $c(x)-k$; converting that budget into an explicit approximation guarantee remains open.

eq:fap-excess:

∑v∈V(x(δ(v))−2)=2(∣F0∣+c(x)−n)=2(c(x)−k).\sum_{v\in V}\bigl(x(\delta(v))-2\bigr) =2\bigl(|F_0|+c(x)-n\bigr) =2\bigl(c(x)-k\bigr).

eq:fap-stability:

∑v∈V(dF0(v)−2)++∑v: dF0(v)=2x(δ(v)∩H)≤2(c(x)−k).\sum_{v\in V}\bigl(d_{F_0}(v)-2\bigr)_{+} +\sum_{v:\,d_{F_0}(v)=2}x(\delta(v)\cap H) \le 2\bigl(c(x)-k\bigr).

— A $59/33$ Cut-LP Guarantee for Matching Augmentation  (2609.26531 - Alimi et al., 22 Sep 2026) in Section Discussion