Beating the 1/2 approximation with deterministic nearly-linear independence-oracle queries

Establish whether a deterministic matroid intersection algorithm exists that uses a nearly linear number of independence-oracle queries and achieves an approximation ratio strictly greater than 1/2 for almost the entire range of values of r, where r denotes the cardinality of a maximum common independent set.

Background

A maximal common independent set, obtainable by a simple greedy algorithm in linear time, guarantees a 1/2-approximation for matroid intersection. Despite this baseline, achieving any approximation ratio strictly larger than 1/2 deterministically with nearly-linear independence-oracle queries has remained elusive.

Prior randomized work using O(n) independence queries achieved (1/2 + δ) for some constant δ > 0, but a deterministic improvement over 1/2 with nearly-linear queries is unknown. This paper provides a deterministic (2/3 − ε)-approximation using nearly-linear queries, addressing this gap for certain regimes.