Stronger Approximation Guarantees for Non-Monotone γ-Weakly DR-Submodular Maximization

Abstract

We study the maximization of nonnegative, non-monotone γ-weakly diminishing returns (DR) submodular functions over down-closed convex bodies. The weakly DR model relaxes classical diminishing returns by allowing marginal gains to decay up to a multiplicative factor γ ∈ (0, 1], capturing a broad class of objectives that interpolate between monotone and fully non-monotone DR submodularity. Existing methods in this regime achieve guarantees that deteriorate rapidly as γ decreases and fail to recover the best known bounds in the fully DR case. We develop a γ-aware algorithmic framework that combines a Frank-Wolfe guided measured continuous greedy procedure with a γ-weighted double-greedy method. Our analysis explicitly accounts for the asymmetric structure induced by weak diminishing returns, yielding γ-dependent progress certificates that remain valid across the entire weakly DR spectrum. As a result, we obtain an approximation guarantee that strictly improves upon the baseline γe−γ for all γ ∈ (0, 1) and recovers the current best constant 0.401 when γ = 1. The proposed algorithms are projection free, use only first order information and linear optimization oracles, and run in polynomial time. © 2026, Transactions on Machine Learning Research. All rights reserved.