[Paper Review] Submodular Maximization under Supermodular Constraint: Greedy Guarantees

Ajitesh Srivastava, Shanghua Teng|arXiv (Cornell University)|Feb 18, 2026

Complexity and Algorithms in Graphs0 citations

TL;DR

The paper analyzes maximizing a monotone submodular function under an upper-bound supermodular constraint and proposes a ratio-based greedy algorithm with bicriteria guarantees, plus duality results and experiments.

ABSTRACT

Motivated by a wide range of applications in data mining and machine learning, we consider the problem of maximizing a submodular function subject to supermodular cost constraints. In contrast to the well-understood setting of cardinality and matroid constraints, where greedy algorithms admit strong guarantees, the supermodular constraint regime remains poorly understood -- guarantees for greedy methods and other efficient algorithmic paradigms are largely open. We study this family of fundamental optimization problems under an upper-bound constraint on a supermodular cost function with curvature parameter $γ$. Our notion of supermodular curvature is less restrictive than prior definitions, substantially expanding the class of admissible cost functions. We show that our greedy algorithm that iteratively includes elements maximizing the ratio of the objective and constraint functions, achieves a $\left(1 - e^{-(1-γ)} ight)$-approximation before stopping. We prove that this approximation is indeed tight for this algorithm. Further, if the objective function has a submodular curvature $c$, then we show that the bound further improves to $\left(1 - (1- (1-c)(1-γ))^{1/(1-c)} ight)$, which can be further improved by continuing to violate the constraint. Finally, we show that the Greedy-Ratio-Marginal in conjunction with binary search leads to a bicriteria approximation for the dual problem -- minimizing a supermodular function under a lower bound constraint on a submodular function. We conduct a number of experiments on a simulation of LLM agents debating over multiple rounds -- the task is to select a subset of agents to maximize correctly answered questions. Our algorithm outperforms all other greedy heuristics, and on smaller problems, it achieves the same performance as the optimal set found by exhaustive search.

Motivation & Objective

• Motivate and formalize the problem of maximizing a submodular function subject to a supermodular cost upper bound.
• Introduce a greedy ratio-based algorithm and establish bicriteria approximation guarantees tied to supermodular curvature.
• Analyze how curvature of the objective and constraint improves guarantees and extend results to the dual problem.
• Demonstrate practical performance via simulations of LLM debating agents and compare with other heuristics.

Proposed method

• Define the SMSC problem with f monotone submodular and g monotone supermodular and an upper budget theta.
• Propose a Greedy Ratio-Marginal rule that selects the element maximizing f(v|S)/g(v|S) and stops when the budget is exceeded.
• Introduce a curvature-based analysis with gamma for g and derive bicriteria guarantees of the form f(S_k) ≥ (1 − e^{−(1−γ)}) f(S*) and g(S_k) ≤ ((2−γ)/(1−γ)) θ.
• Provide refinements when the curvature of f is known to tighten the guarantees.
• Extend the analysis to continued overflow beyond the constraint and obtain improved bounds.
• Describe a dual problem approach using binary search to convert bicriteria primal guarantees into dual guarantees.

Figure 1 . Summary of our results for submodular maximization with supermodular constraint: We find the approximation factor as a function of the supermodular curvature (right) when Greedy algorithm exceeds the constraint for the first time. We obtain better approximation when submodular objective h

Experimental results

Research questions

• RQ1 What approximation guarantees can greedy methods achieve for maximizing a submodular function under a supermodular constraint?
• RQ2 How does supermodular curvature γ and submodular curvature c affect the performance of greedy algorithms in SMSC?
• RQ3 Can the greedy approach be extended to provide bicriteria guarantees when the constraint is violated, and how tight are these bounds?
• RQ4 How can the dual problem (minimizing a supermodular function under a submodular constraint) be addressed using the proposed methods?
• RQ5 Do empirical results on realistic simulations (e.g., LLM debating agents) support the theoretical guarantees and competitiveness of the proposed method?

Key findings

• The Greedy Ratio-Marginal algorithm achieves a bicriteria guarantee: after first overflow, either f(S_k) ≥ (1−e^{−(1−γ)}) f(S*) or, with known curvature, the bound improves accordingly.
• The overflow is bounded by β ≤ (2−γ)/(1−γ), yielding a concrete budget violation while preserving a constant-factor approximation.
• If the submodular objective has curvature c, the bound improves to a tighter expression that depends on c and γ.
• The authors prove the tightness of the greedy bound, showing there exist instances where the approximation approaches 1−e^{−(1−γ)}.
• A duality result shows a binary-search based reduction can convert bicriteria guarantees for the primal into bicriteria guarantees for the dual problem.

Figure 2 . Instance of max cover with supermodular cost.