derive submodular bounds

This study addresses the problem of maximizing a submodular function subject to a supermodular cost constraint. The authors propose a greedy ratio marginal algorithm that iteratively selects the element with the highest ratio of marginal gain in the objective to marginal cost. A key innovation is the introduction of a relaxed notion of supermodular curvature, parameterized by γ, which broadens the class of admissible cost functions. For this setting, the work establishes the first tight approximation ratio guarantee for the greedy algorithm. By further incorporating the curvature c of the submodular objective, the theoretical bound is refined, and the analysis is extended to yield a bicriteria approximation for the dual problem. Empirical evaluations on multi-round LLM agent debate simulation tasks demonstrate that the proposed method outperforms existing greedy heuristics and achieves optimal performance on small-scale instances via exhaustive search.

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 studies the minimization and maximization of submodular/supermodular set functions $f(S)/|S|$ over nonempty subsets, unifying classical problems such as densest subgraph discovery, densest supermodular set detection, and submodular function minimization. We establish, for the first time, a strong polynomial-time equivalence between these ratio optimization problems and the minimum-norm point (MNP) problem. To handle non-monotone and negative-valued functions, we propose two novel frameworks: Unconstrained Submodular Ratio Optimization (USSS) and Unconstrained Supermodular Ratio Optimization (UDSS). The theoretical foundation integrates MNP optimization over base polyhedra, the Fujishige–Wolfe algorithm, and the SuperGreedy++ heuristic. Extensive experiments across 400+ benchmarks demonstrate that our generic convex-optimization and network-flow approaches consistently outperform task-specific baselines, achieving scalable, state-of-the-art performance on large-scale real-world and synthetic datasets.

This paper addresses the monotone submodular maximization problem subject to a knapsack constraint (Submodular Knapsack Problem), aiming to deliver verifiably optimal solutions for practical applications. We propose the first dedicated branch-and-bound framework for this problem, integrating three key innovations: (i) tight upper-bound estimation leveraging submodularity, (ii) pruning rules derived directly from submodular properties, and (iii) a dynamic variable ordering strategy. Evaluated on three benchmark instance classes, our method significantly outperforms both general-purpose integer programming solvers and state-of-the-art heuristics in efficiency and scalability: it achieves an average 3.2× speedup on medium-scale instances and, for the first time, solves several previously intractable instances to optimality in polynomial time. Theoretical analysis ensures solution correctness and bound tightness, while empirical results demonstrate robust performance across diverse problem scales—establishing a new standard for exact algorithms in submodular optimization with practical relevance.

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 addresses the challenge of evaluating solution quality for submodular maximization under a knapsack constraint, where existing algorithms offer only conservative worst-case approximation guarantees that poorly reflect practical performance. The paper introduces, for the first time, a data-dependent upper bound that is theoretically guaranteed to be strictly tighter than the trivial bound on the optimal value. By integrating submodular optimization theory with data-driven techniques, the authors develop a novel upper-bound estimation framework tailored to knapsack-constrained settings. Extensive experiments on multiple real-world datasets demonstrate that the proposed bound provides a significantly tighter approximation of the true optimum, thereby substantially enhancing the certifiable quality of obtained solutions and markedly outperforming conventional worst-case bounds.

This work addresses the property testing of $k$-submodular functions, a high-dimensional generalization of submodularity defined over partial partitions of a ground set that simultaneously satisfies diminishing marginal returns and pairwise monotonicity constraints. Accounting for structural differences under $\ell_p$ and Hamming distances, the paper introduces two types of local refutation patterns—violating squares and triangles—and analyzes the combinatorial barriers arising from their conflicting repair requirements. Leveraging techniques including implicit learning over hypergrids, partial-partition filters, ideal repairs, pseudo-DNF representations, and product-domain learning, the authors construct a constant-query, non-adaptive, one-sided tester under $\ell_p$ distance. For the Hamming distance, they design subexponential-query testers for two constituent subproperties and present the first adaptive tester for monotone $k$-submodularity with bounded value ranges.

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.