Nearly-linear running time for all matroid ranks

Develop randomized or deterministic algorithms for maximizing a non-negative monotone submodular function f: 2^N → R subject to a matroid constraint M = (N, I) that achieve nearly-linear running time (i.e., tilde O(n) query complexity) for all values of the matroid rank r, improving beyond current results that are nearly-linear only in restricted regimes (e.g., r = tilde O(n^{2/3})).

Background

The paper presents a deterministic non-oblivious local search algorithm for maximizing a non-negative monotone submodular function under a matroid constraint with approximation 1 − 1/e − ε and deterministic query complexity tilde O(nr), improved to tilde O(n + r√n) using randomization. These complexities are nearly-linear only for certain ranges of r (e.g., r = tilde O(√n) for the randomized variant).

The authors highlight that achieving nearly-linear time across all values of r remains unresolved. They note a recent algorithm [26] that attains tilde O(n + r^{3/2}) query complexity, which is nearly-linear when r = tilde O(n^{2/3}), leaving open the general case for arbitrary r.