Reduce or eliminate exponential dependence on 1/ε in query complexity

Ascertain whether the exponential dependence on 1/ε in the query complexity of algorithms for maximizing a non-negative monotone submodular function subject to a matroid constraint—arising because evaluating the auxiliary function g′ requires O(2^{1/ε}) value-oracle queries to f—can be reduced or avoided.

Background

In Section 4 the authors introduce an auxiliary function g′ defined over an expanded ground set to enable their non-oblivious local search framework. Evaluating g′ requires summing over all subsets J ⊆ [ℓ], with ℓ ≈ 1 + ⌈1/ε⌉, which incurs O(2^{ℓ}) = O(2^{1/ε}) value-oracle calls to f per evaluation.

This evaluation cost induces an exponential dependence on 1/ε in the overall query complexity of their algorithms. The authors point out that even recent related algorithms [26] exhibit the same exponential dependence, emphasizing the unresolved nature of achieving polynomial (or otherwise improved) dependence on 1/ε.