Sub-2 approximation for weighted three-matroid intersection

Establish whether weighted three-matroid intersection admits a polynomial-time approximation factor strictly below 2.

Background

The natural LP for weighted intersection of three matroids has integrality gap exactly 2, and the cited iterative-refinement algorithm gives a weighted LP-relative 2-approximation. The paper records that this LP-relative guarantee is tight, but distinguishes it from approximation relative to the integral optimum.

Whether the underlying weighted optimization problem can nevertheless be approximated with a factor below 2 is explicitly left unresolved. This question concerns algorithmic approximation and is not settled by the paper’s integrality-gap result.

References

It remains open whether weighted $3$-matroid intersection admits an approximation factor below $2$ .

— Integrality-Gap Bounds for Weighted Matchoids and Matroid Intersection  (2609.21477 - Cong et al., 18 Sep 2026) in Section 1, paragraph beginning “For two arbitrary matroids…” in “Previous results.”