Integrality-gap conjecture for weighted k-matroid intersection

Determine whether the natural linear-programming relaxation for weighted intersection of k matroids has integrality gap at most k-1 for every integer k, particularly for k greater than or equal to 4.

Background

The paper studies the natural LP relaxation for selecting a maximum-weight set that is independent in each of k matroids on a common ground set. The conjectured optimal upper bound on its integrality gap is k-1, a value already known for k less than or equal to 3.

For k greater than or equal to 4, the paper improves the previously known general upper bound k to k-1+1/k, but does not establish the conjectured k-1 bound. Thus, determining whether the natural LP always has gap at most k-1 remains unresolved in the range addressed by the new result.

References

It is conjectured that this LP has integrality gap at most $k-1$. The conjecture is known for $k\le3$, but for $k\ge4$ the best general upper bound was $k$.

— Integrality-Gap Bounds for Weighted Matchoids and Matroid Intersection  (2609.21477 - Cong et al., 18 Sep 2026) in Abstract; Section 1, paragraph beginning “Aharoni and Berger proposed…”