PTAS for Matroid Intersection Cover

Establish whether the Matroid Intersection Cover (MIC) problem admits a polynomial-time approximation scheme under its covering constraint.

Background

The paper develops an efficient polynomial-time approximation scheme (EPTAS) for matroid optimization with one linear constraint under a single matroid, but proves that MIC—the problem of minimizing one nonnegative weight subject to a lower-bound covering constraint and feasibility in the intersection of two matroids—admits no EPTAS unless W[1]=FPT. This hardness result leaves open the weaker approximation guarantee of a PTAS for MIC.

The authors identify this question as part of the broader task of characterizing the approximability landscape of the class of matroid optimization problems with linear constraints when feasibility is defined by matroid intersection. The unresolved issue is therefore whether MIC has a PTAS despite the conditional nonexistence of an EPTAS.