Determine whether exact linear matroid intersection is in P

Ascertain whether the exact linear matroid intersection decision problem—given two linear matroids M1=(S, I1) and M2=(S, I2) over the same ground set with integer weights w:S→Z and a target integer T, decide if there exists I∈I1∩I2 such that w(I)=T—admits a deterministic polynomial-time algorithm (i.e., belongs to P).

Background

Exact matching (deciding existence of a perfect matching of exactly a given weight) is a classic problem known to have randomized polynomial-time algorithms but lacking known deterministic polynomial-time algorithms; exact linear matroid intersection generalizes this setting.

The authors note that exact linear matroid intersection has a randomized polynomial-time algorithm but no known deterministic polynomial-time algorithm. Establishing membership in P would represent significant progress and strengthen barriers regarding possible inclusions between CL and NC.