# Algorithms for Approximate Minimization of the Difference Between Submodular Functions, with Applications

arXiv: https://arxiv.org/abs/1408.2051 · HTML: https://mlsanity.com/paper/1408.2051/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Aug 2014.

## Built on by (most cited first, 17 of 17)

- [Co-Mixup: Saliency Guided Joint Mixup with Supermodular Diversity](https://mlsanity.com/paper/2102.03065/) (2021)
- [Submodularity In Machine Learning and Artificial Intelligence](https://mlsanity.com/paper/2202.00132/) (2022)
- [Global Reinforcement Learning: Beyond Linear and Convex Rewards via Submodular Semi-gradient Methods](https://mlsanity.com/paper/2407.09905/) (2024)
- [Training Data Subset Selection for Regression with Controlled Generalization Error](https://mlsanity.com/paper/2106.12491/) (2021)
- [Learning Interpretable Decision Rule Sets: A Submodular Optimization Approach](https://mlsanity.com/paper/2206.03718/) (2022)
- [Decomposition Polyhedra of Piecewise Linear Functions](https://mlsanity.com/paper/2410.04907/) (2024)
- [ktexttt-experts -- Online Policies and Fundamental Limits](https://mlsanity.com/paper/2110.07881/) (2021)
- [Difference of Submodular Minimization via DC Programming](https://mlsanity.com/paper/2305.11046/) (2023)
- [Balancing Utility and Fairness in Submodular Maximization (Technical Report)](https://mlsanity.com/paper/2211.00980/) (2022)
- [Nonlinear Bayesian optimal experimental design using logarithmic Sobolev inequalities](https://mlsanity.com/paper/2402.15053/) (2024)
- [Online Nonsubmodular Minimization with Delayed Costs: From Full Information to Bandit Feedback](https://mlsanity.com/paper/2205.07217/) (2022)
- [Non-approximate Inference for Collective Graphical Models on Path Graphs via Discrete Difference of Convex Algorithm](https://mlsanity.com/paper/2102.09191/) (2021)
- [A Resilience Framework for Bi-Criteria Combinatorial Optimization with Bandit Feedback](https://mlsanity.com/paper/2503.12285/) (2025)
- [Inexact Column Generation for Bayesian Network Structure Learning via Difference-of-Submodular Optimization](https://mlsanity.com/paper/2505.11089/) (2025)
- [Discrete and Continuous Difference of Submodular Minimization](https://mlsanity.com/paper/2506.07952/) (2025)
- [Redundancy Meets Synergy: Dependency-aware Expert Selection for MoE via Submodular Optimization](https://mlsanity.com/paper/2610.00558/) (2026)
- [Generalized Determinantal Point Processes for Attraction and Repulsion with Distinct Interaction Kernels](https://mlsanity.com/paper/2610.12894/) (2026)

## Abstract (arXiv)

We extend the work of Narasimhan and Bilmes [30] for minimizing set functions representable as a dierence between submodular functions. Similar to [30], our new algorithms are guaranteed to monotonically reduce the objective function at every step. We empirically and theoretically show that the per-iteration cost of our algorithms is much less than [30], and our algorithms can be used to efficiently minimize a dierence between submodular functions under various combinatorial constraints, a problem not previously addressed. We provide computational bounds and a hardness result on the multiplicative inapproximability of minimizing the dierence between submodular functions. We show, however, that it is possible to give worst-case additive bounds by providing a polynomial time computable lower-bound on the minima. Finally we show how a number of machine learning problems can be modeled as minimizing the dierence between submodular functions. We experimentally show the validity of our algorithms by testing them on the problem of feature selection with submodular cost features.