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

2025-05-23

AI Summary

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.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Topic discovery and tracking
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

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 o 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 extsc{SuperGreedy++} act as universal solvers for all these problems, including sub-modular function minimization. Theoretically, we explain why extsc{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.

Problem

• Optimizing average value of submodular/supermodular set functions
• Generalizing classical problems like DSG, DSS, and SFM
• Exploring algorithmic equivalence and universal solvers for these problems

Innovation

• Equivalent polynomial-time reductions for multiple problems
• Universal solvers like SuperGreedy++ and Fujishige-Wolfe
• General-purpose convex and flow-based optimization techniques