Stronger Approximation Guarantees for Non-Monotone $\gamma$-Weakly DR-Submodular Maximization

TMLR Paper6966 Authors

Abstract:

We study the maximization of nonnegative, non-monotone $\gamma$-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 $\gamma \in (0,1]$, 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 $\gamma$ decreases and fail to recover the best known bounds in the fully DR case.

We develop a $\gamma$-aware algorithmic framework that combines a Frank--Wolfe guided measured continuous greedy procedure with a $\gamma$-weighted double-greedy method. Our analysis explicitly accounts for the asymmetric structure induced by weak diminishing returns, yielding $\gamma$-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 $\gamma e^{-\gamma}$ for all $\gamma \in (0,1)$ and recovers the current best constant $0.401$ when $\gamma = 1$. The proposed algorithms are projection-free, use only first-order information and linear optimization oracles, and run in polynomial time.

Submission Type: Long submission (more than 12 pages of main content)

Assigned Action Editor: ~Zheng_Wen1

Submission Number: 6966