[论文解读] Submodular Maximization under Supermodular Constraint: Greedy Guarantees

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

Complexity and Algorithms in Graphs被引用 0

一句话总结

该论文分析在上限超模约束下最大化单调子模函数，并提出基于比率的贪婪算法及双目标保证，以及对偶性结果和实验。

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.

研究动机与目标

• 推动并形式化在超模成本上界约束下最大化子模函数的问题。
• 引入基于贪婪比率的算法并建立与超模曲率相关的双目标近似保证。
• 分析目标与约束的曲率如何提升保证，并将结果扩展到对偶问题。
• 通过对LLM辩论代理的仿真演示实际性能，并与其他启发式方法进行比较。

提出的方法

• 将SMSC问题定义为f为单调子模、g为单调超模且存在上界预算theta。
• 提出Greedy Ratio-Marginal规则，选择使 f(v|S)/g(v|S) 最大的元素，并在预算超出时停止。
• 引入基于曲率的分析，记g的伽马为gamma，推导形如 f(S_k) ≥ (1 − e^{−(1−γ)}) f(S*) 与 g(S_k) ≤ ((2−γ)/(1−γ)) θ 的双目标保证。
• 在已知f的曲率时给出改进以收紧保证。
• 将分析扩展到对约束的持续溢出并获得改进界。
• 描述使用二分查找的对偶问题方法，将双目标原始保证转化为对偶保证。

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

实验结果

研究问题

• RQ1 贪婪方法在在超模约束下最大化子模函数时能达到怎样的近似保证？
• RQ2 超模曲率γ与子模曲率c如何影响SMSC中贪婪算法的性能？
• RQ3 当约束被违反时，贪婪方法能否扩展为提供双目标保证？这些界限有多紧？
• RQ4 如何使用所提方法处理对偶问题（在子模约束下最小化超模函数）？
• RQ5 在现实仿真（如LLM辩论代理）中的经验结果是否支持理论保证及方法的竞争力？

主要发现

• Greedy Ratio-Marginal算法实现了双目标保证：首次溢出后，要么 f(S_k) ≥ (1−e^{−(1−γ)}) f(S*)，若已知曲率，则上界相应改进。
• 溢出由 β ≤ (2−γ)/(1−γ) 约束，在保持常数因子近似的同时实现具体的预算违反。
• 若子模目标具有曲率c，界限对c和γ的依赖得到更紧的表达。
• 作者证明贪婪界限的紧性，存在实例使近似度趋近于1−e^{−(1−γ)}。
• 一个对偶性结果表明，基于二分搜索的约简可以将原问题的双目标保证转化为对偶问题的双目标保证。

Figure 2 . Instance of max cover with supermodular cost.