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

TL;DR

The paper addresses ratio optimization for submodular and supermodular set functions, unifying classic problems such as Densest Subgraph, Submodular Function Minimization, and their ratio variants by introducing Unrestricted Sparsest Submodular Set (USSS) and Unrestricted Densest Supermodular Set (UDSS). It establishes strong-polynomial reductions showing equivalence among five problems (SFM, DSS, MNP, USSS, UDSS) and demonstrates that solutions to the Minimum Norm Point problem on the base polytope yield additive approximations across all these problems. The authors position SuperGreedy++ and Wolfe's MNP algorithm as universal solvers within a Frank-Wolfe framework, connecting Fujishige’s theory with dense-decomposition and validating the approach with extensive experiments on large-scale real and synthetic data. Empirically, general-purpose convex and flow-based methods often outperform problem-specific baselines, highlighting a scalable and unified optimization pathway for submodular and supermodular ratio problems with practical impact on tasks like heavy-node detection and minimum cut. The work provides both theoretical reductions and practical evidence that broad optimization techniques can robustly solve a wide class of ratio problems in submodular optimization.

Abstract

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.

Figures (31)

• Figure 1: Performance of algorithms across selected HNSN and Minimum $s$-$t$ Cut instances.
• Figure 2: HNSN results across 12 datasets.
• Figure 3: Min-Cut vs Wall Clock Time (Part 1).
• Figure 4: Min-Cut vs Wall Clock Time (Part 2).
• Figure 5: DSG density over time

Theorems & Definitions (16)

• Definition 9: Base Polymatroid
• Definition 10: Base Contrapolymatroid
• Lemma 11
• Theorem 12
• Theorem 13
• Theorem 14
• Definition 15: Curvature constant
• Definition 16: Approximate LMO
• Lemma 17: pmlr-v28-jaggi13
• Lemma 18