FPTAS for Graphic and Explicitly Represented Linear Matroids

Determine whether a fully polynomial-time approximation scheme exists for the matroid optimization problems with a linear constraint considered in the paper when the underlying matroid is graphic or is a linear matroid given by an explicit representation.

Background

The paper establishes an EPTAS for all nontrivial variants of matroid optimization with a single linear constraint and notes that a fully polynomial-time approximation scheme (FPTAS) is impossible for general instances under the cited complexity results. However, the paper's EPTAS framework applies to special representations including graphic matroids and explicitly represented linear matroids, leaving open whether these structured settings permit the stronger FPTAS guarantee.

The question concerns the existence of stronger approximation guarantees in these specific matroid classes; it is not resolved by the general lower bound for arbitrary matroid instances.

References

Finally, it would be interesting to identify settings in which stronger guarantees are attainable. For instance, for graphic matroids and for linear matroids given by an explicit representation, we obtain an EPTAS, while the existence of an FPTAS in these cases remains open.

— Tight Approximation Results for Matroid Optimization with a Linear Constraint  (2609.28708 - Doron-Arad et al., 23 Sep 2026) in Section Discussion