Deterministic nearly-linear-query (1 − ε)-approximation for matroid intersection

Determine whether there exists a deterministic algorithm for the matroid intersection problem that, for any ε > 0 and for instances with maximum common independent set cardinality r across almost the entire range of r (beyond the case r = Θ(n)), uses Õε(n) independence-oracle queries to compute a (1 − ε)-approximate common independent set.

Background

Recent advances by Quanrud and by Blikstad–Tu have produced randomized algorithms that achieve (1 − ε)-approximation for matroid intersection with nearly-linear independence-oracle queries. However, these results do not provide deterministic guarantees across general values of r.

The only known deterministic nearly-linear-query (1 − ε)-approximation currently applies when r = Θ(n) (with ε constant). This leaves open whether a deterministic Õε(n)-query algorithm achieving (1 − ε) approximation exists for broader regimes of r. The present paper offers a deterministic (2/3 − ε)-approximation as a step toward this goal.

References

Both the recent $(1 - )$-approximation algorithms by Quanrud and Blikstad--Tu are randomized. Therefore, it remains an open question whether a deterministic $\tilde{O}_{}(n)$ independence-query $(1 - )$-approximation algorithm can be achieved for almost the entire range of $r$.

— Deterministic $(2/3-\varepsilon)$-Approximation of Matroid Intersection Using Nearly-Linear Independence-Oracle Queries (2410.18820 - Terao, 24 Oct 2024) in Section 1 (Introduction)