# One-for-many Counterfactual Explanations by Column Generation

arXiv: https://arxiv.org/abs/2402.09473 · HTML: https://mlsanity.com/paper/2402.09473/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 36 references that are in the map.
- First posted on arXiv 12 Feb 2024.

## Builds on (most cited first, 16 of 16)

- [Explanation in Artificial Intelligence: Insights from the Social Sciences](https://mlsanity.com/paper/1706.07269/) (2017)
- [Counterfactual Explanations without Opening the Black Box: Automated Decisions and the GDPR](https://mlsanity.com/paper/1711.00399/) (2017)
- [Explaining Machine Learning Classifiers through Diverse Counterfactual Explanations](https://mlsanity.com/paper/1905.07697/) (2019)
- [Interpretable Machine Learning: Fundamental Principles and 10 Grand Challenges](https://mlsanity.com/paper/2103.11251/) (2021)
- [Actionable Recourse in Linear Classification](https://mlsanity.com/paper/1809.06514/) (2018)
- [Techniques for Interpretable Machine Learning](https://mlsanity.com/paper/1808.00033/) (2018)
- [Model-Agnostic Counterfactual Explanations for Consequential Decisions](https://mlsanity.com/paper/1905.11190/) (2019)
- [Interpretable Machine Learning -- A Brief History, State-of-the-Art and Challenges](https://mlsanity.com/paper/2010.09337/) (2020)
- [Good Counterfactuals and Where to Find Them: A Case-Based Technique for Generating Counterfactuals for Explainable AI (XAI)](https://mlsanity.com/paper/2005.13997/) (2020)
- [Interpretable Classification Models for Recidivism Prediction](https://mlsanity.com/paper/1503.07810/) (2015)
- [Optimal Counterfactual Explanations in Tree Ensembles](https://mlsanity.com/paper/2106.06631/) (2021)
- [On the computation of counterfactual explanations -- A survey](https://mlsanity.com/paper/1911.07749/) (2019)
- [JANOS: An Integrated Predictive and Prescriptive Modeling Framework](https://mlsanity.com/paper/1911.09461/) (2019)
- [GLOBE-CE: A Translation-Based Approach for Global Counterfactual Explanations](https://mlsanity.com/paper/2305.17021/) (2023)
- [Explaining Groups of Instances Counterfactually for XAI: A Use Case, Algorithm and User Study for Group-Counterfactuals](https://mlsanity.com/paper/2303.09297/) (2023)
- [Generating collective counterfactual explanations in score-based classification via mathematical optimization](https://mlsanity.com/paper/2310.12822/) (2023)

## Built on by (most cited first, 2 of 2)

- [Understanding Fixed Predictions via Confined Regions](https://mlsanity.com/paper/2502.16380/) (2025)
- [Trustworthy Machine Learning through the Lens of Combinatorial Optimization: Survey and Research Perspectives](https://mlsanity.com/paper/2607.07762/) (2026)

## Abstract (arXiv)

In this paper, we consider the problem of generating a set of counterfactual explanations for a group of instances, with the one-for-many allocation rule, where one explanation is allocated to a subgroup of the instances. For the first time, we solve the problem of minimizing the number of explanations needed to explain all the instances, while considering sparsity by limiting the number of features allowed to be changed collectively in each explanation. A novel column generation framework is developed to efficiently search for the explanations. Our framework can be applied to any black-box classifier, like neural networks. Compared with a simple adaptation of a mixed-integer programming formulation from the literature, the column generation framework dominates in terms of scalability, computational performance and quality of the solutions.