Enhanced Deterministic Approximation Algorithm for Non-monotone Submodular Maximization under Knapsack Constraint with Linear Query Complexity
Canh V. Pham
TL;DR
This work addresses SMK for a ground set with a non-monotone submodular objective by introducing a deterministic algorithm, $\ extsf{EDL}$, that attains a $5+\epsilon$-approximation while preserving near-linear query complexity $O(n \log(1/\epsilon)/\epsilon)$. The method combines a threshold-greedy scheme with two disjoint candidate sets and leverages an initial feasible solution from the LA subroutine to narrow the search range for the optimum. Theoretical analysis connects the cost structure of candidate solutions to the optimum, enabling a tight bound of $f(\mathsf{O})\le(5+\epsilon)f(S)$ and hence a significant improvement over the prior deterministic $6+\epsilon$ factor. The results advance deterministic SMK optimization by delivering stronger performance guarantees without increasing query effort, with practical implications for data summarization, influence propagation, and related combinatorial optimization tasks.
Abstract
In this work, we consider the Submodular Maximization under Knapsack (SMK) constraint problem over the ground set of size $n$. The problem recently attracted a lot of attention due to its applications in various domains of combination optimization, artificial intelligence, and machine learning. We improve the approximation factor of the fastest deterministic algorithm from $6+ε$ to $5+ε$ while keeping the best query complexity of $O(n)$, where $ε>0$ is a constant parameter. Our technique is based on optimizing the performance of two components: the threshold greedy subroutine and the building of two disjoint sets as candidate solutions. Besides, by carefully analyzing the cost of candidate solutions, we obtain a tighter approximation factor.