Bridging the Gap Between General and Down-Closed Convex Sets in Submodular Maximization

• URL: http://arxiv.org/abs/2401.09251v1
• Date: Wed, 17 Jan 2024 14:56:42 GMT
• Title: Bridging the Gap Between General and Down-Closed Convex Sets in Submodular Maximization
• Authors: Loay Mualem, Murad Tukan, Moran Fledman
• Abstract summary: We show that Mualem citemualem23re shows that this approach cannot smooth between down- and non-down-closed constraints. In this work, we suggest novel offline and online algorithms based on a natural decomposition of the body into two distinct convex bodies. We also empirically demonstrate the superiority of our proposed algorithms across three offline and two online applications.
• Score: 8.225819874406238
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Optimization of DR-submodular functions has experienced a notable surge in significance in recent times, marking a pivotal development within the domain of non-convex optimization. Motivated by real-world scenarios, some recent works have delved into the maximization of non-monotone DR-submodular functions over general (not necessarily down-closed) convex set constraints. Up to this point, these works have all used the minimum $\ell_\infty$ norm of any feasible solution as a parameter. Unfortunately, a recent hardness result due to Mualem \& Feldman~\cite{mualem2023resolving} shows that this approach cannot yield a smooth interpolation between down-closed and non-down-closed constraints. In this work, we suggest novel offline and online algorithms that provably provide such an interpolation based on a natural decomposition of the convex body constraint into two distinct convex bodies: a down-closed convex body and a general convex body. We also empirically demonstrate the superiority of our proposed algorithms across three offline and two online applications.

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