Corporate Needs You to Find the Difference: Revisiting Submodular and Supermodular Ratio Optimization Problems

We study the problem of minimizing or maximizing the average value $ f(S)/|S| $ of a submodular or supermodular set function $ f: 2^V \to \mathbb{R} $ over non-empty subsets $ S \subseteq V $. This generalizes classical problems such as Densest Subgraph (DSG), Densest Supermodular Set (DSS), and Submodular Function Minimization (SFM). Motivated by recent applications, we introduce two broad formulations: Unrestricted Sparsest Submodular Set (USSS) and Unrestricted Densest Supermodular Set (UDSS), which allow for negative and non-monotone functions. We show that DSS, SFM, USSS, UDSS, and the Minimum Norm Point (MNP) problem are equivalent under strongly polynomial-time reductions, enabling algorithmic crossover. In particular, viewing these through the lens of the MNP in the base polyhedron, we connect Fujishige's theory with dense decomposition, and show that both Fujishige-Wolfe's algorithm and the heuristic \textsc{SuperGreedy++} act as universal solvers for all these problems, including sub-modular function minimization. Theoretically, we explain why \textsc{SuperGreedy++} is effective beyond DSS, including for tasks like submodular minimization and minimum $ s $-$ t $ cut. Empirically, we test several solvers, including the Fujishige-Wolfe algorithm on over 400 experiments across seven problem types and large-scale real/synthetic datasets. Surprisingly, general-purpose convex and flow-based methods outperform task-specific baselines, demonstrating that with the right framing, general optimization techniques can be both scalable and state-of-the-art for submodular and supermodular ratio problems.