maximize submodular functions

This paper investigates the fundamental gap in approximation performance between the maximum coverage problem and general monotone submodular maximization under a cardinality constraint equal to a fixed fraction $c$ of the ground set size. Contrary to the conventional belief that both problems share identical approximation limits, we establish the first provable separation: when $c = 1/2$, maximum coverage admits a $0.7533$-approximation—strictly exceeding the best possible approximation ratio for arbitrary monotone submodular functions. Technically, we derive a tight approximation bound $1 - (1 - c)^{1/c}$ for the submodular case and provide a matching hardness lower bound, particularly tight when $c = 1/s$ for integer $s$. Our analysis integrates constructive hardness proofs, an improved greedy algorithm, and refined perturbation arguments. These results collectively demonstrate an intrinsic divergence in the optimal approximation ratios of the two problems under proportional cardinality constraints.

This paper studies the budget-constrained $k$-submodular maximization problem, an NP-hard optimization task. We propose the 1-Guess Greedy algorithm: by guessing at most one critical element from the optimal solution in a single pass—and integrating greedy selection with threshold-based truncation—it achieves tight approximation ratios of $1/2$ for monotone and $1/3$ for non-monotone objectives, respectively; this is the first proof of a tight $1/2$-approximation bound under budget constraints. Technically, we introduce a novel continuous analysis framework based on the multilinear extension, which unifies treatment across diverse constraints while ensuring simplicity, parallelizability, and theoretical tightness. The algorithm runs in $ ilde{O}(n^2 k^2)$ time, substantially improving upon prior approaches.

This work addresses the problem of cardinality-constrained multi-objective submodular maximization under differential privacy, where the goal is to select at most $k$ elements from sensitive data to maximize the minimum of $d$ monotone submodular functions. The study introduces differential privacy into this setting for the first time and proposes two novel algorithms: one extends the classical greedy strategy with privacy guarantees, and the other integrates a function truncation technique. Both algorithms provide rigorous theoretical approximation guarantees. Empirical evaluations on maximum coverage and facility location tasks demonstrate that the proposed methods effectively balance utility and privacy, thereby bridging a critical gap in the intersection of differential privacy and multi-objective submodular optimization.

This paper studies the minimum-cost set cover problem under a fixed constant $ r $ of monotone submodular constraints: given a ground set $ N $, a cost function $ c: N o mathbb{R}_+ $, $ r $ monotone submodular functions $ f_i $, and thresholds $ b_i $, find a minimum-cost subset $ S subseteq N $ satisfying $ f_i(S) geq b_i $ for all $ i $. To overcome the bottleneck where classical algorithms’ approximation ratios degrade with $ r $, we propose the first bi-criteria randomized approximation algorithm. Our method integrates LP relaxation, weighted covering function techniques, and structural properties of deletion-closed systems. In expectation, the solution cost is at most $ alpha cdot mathrm{OPT} $, while achieving coverage ratio $ 1 - 1/e^alpha - varepsilon $. For weighted covering functions, we obtain an approximation ratio of $ (1+varepsilon)frac{e}{e-1}(1+eta) $, breaking the $ r $-dependent logarithmic lower bound.

This paper addresses the discrete maximization of non-monotone submodular functions under cardinality and matroid constraints, breaking the long-standing $1/e approx 0.367$ approximation barrier for combinatorial algorithms. We propose the **guided randomized greedy framework**, integrating fast local search while avoiding costly continuous extensions. We further design **deterministic and nearly-linear-time variants** that preserve the approximation guarantees. Under cardinality constraints, our algorithm achieves a $0.385$ approximation ratio—improving upon the previous best $0.367$; under matroid constraints, it attains $0.305$, surpassing $0.281$. The deterministic variant achieves $0.377$ with nearly-linear time complexity. To our knowledge, this is the first purely combinatorial algorithm—requiring no continuous optimization—that strictly exceeds the $1/e$ barrier, significantly enhancing scalability and practical applicability.

This work studies the problem of maximizing submodular functions subject to a knapsack constraint under differential privacy, covering both monotone and non-monotone settings. The authors propose efficient approximation algorithms that achieve an optimal $(1-1/e)$-approximation ratio in the monotone case while significantly reducing additive error and query complexity. For the non-monotone setting, they present the first differentially private algorithm with theoretical guarantees, attaining an expected approximation ratio of $1/4$ and additive error comparable to the best-known monotone algorithms. Their approach integrates differential privacy mechanisms with submodular optimization techniques, knapsack constraint handling, and randomized sampling combined with noise injection.

Large-scale subset selection asks for a small useful set of examples, features, sensors, seed users, or context passages from an enormous ground set. Submodular maximization is a canonical model for such diminishing-returns problems, but rapidly growing datasets make even linear-time algorithms ever costlier. We study \emph{containment pruning}: first reduce the ground set to a smaller core $P$, then require that $P$ contain a near-optimal feasible solution for every downstream budget up to~$k$. Prior work has formulated many heuristics, but the theoretical limits of this preprocessing problem are largely unknown. For monotone submodular objectives, we prove that $1-1/e$ is tight: greedy achieves this containment factor, and no algorithm can beat it even with a larger pruning budget. For non-monotone objectives, we give the first$1/2-\varepsilon$ containment algorithms under cardinality constraints and extend the approach to knapsack constraints. This $1/2$ factor exceeds the best known algorithmic ratio and the known hardness threshold for non-monotone maximization, showing that pruning can be provably easier than optimization. Empirically, pruning lets an exact IP solver run on the reduced MaxCut instance with a ${\approx}620\times$ speedup, and proof-of-concept experiments on LLM context selection demonstrate the utility of non-monotone submodular proxies and our proposed containment algorithms.

This work addresses the maximization of non-monotone submodular functions under both matroid and knapsack constraints. The authors propose a deterministic approximation algorithm based on an extended multilinear extension, which integrates deterministic continuous optimization with efficient discretization techniques. This approach achieves improved approximation ratios for both constraint types simultaneously while maintaining polynomial query complexity—the first such result in the deterministic setting. Specifically, the algorithm attains a (0.385 − ε)-approximation under matroid constraints and a (0.367 − ε)-approximation under knapsack constraints, surpassing the previous best deterministic guarantees of 0.367 and 0.25, respectively, and establishing the current state-of-the-art deterministic performance for these problems.

This work addresses the problem of maximizing a monotone submodular function subject to a matroid independence constraint. The authors introduce, for the first time, a stochastic Poisson process into this domain and propose a novel algorithm that achieves efficient optimization through only a small number of single-element exchanges, without requiring discretization or rounding. The method features a simple structure, circumventing the need for complex rounding procedures inherent in traditional approaches, while attaining the tight $(1-1/e)$ approximation guarantee. As applications, the framework effectively solves submodular welfare maximization as well as general and separable assignment problems, yielding significant improvements in computational efficiency.