# Ordered Counterfactual Explanation by Mixed-Integer Linear Optimization

arXiv: https://arxiv.org/abs/2012.11782 · HTML: https://mlsanity.com/paper/2012.11782/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Dec 2020.

## Built on by (most cited first, 11 of 11)

- [Data-centric Artificial Intelligence: A Survey](https://mlsanity.com/paper/2303.10158/) (2023)
- [Robust Counterfactual Explanations for Tree-Based Ensembles](https://mlsanity.com/paper/2207.02739/) (2022)
- [When Deep Learning Meets Polyhedral Theory: A Survey](https://mlsanity.com/paper/2305.00241/) (2023)
- [Optimal Counterfactual Explanations in Tree Ensembles](https://mlsanity.com/paper/2106.06631/) (2021)
- [Privacy Implications of Explainable AI in Data-Driven Systems](https://mlsanity.com/paper/2406.15789/) (2024)
- [DiCoFlex: Model-agnostic diverse counterfactuals with flexible control](https://mlsanity.com/paper/2505.23700/) (2025)
- [Counterfactual Maps: What They Are and How to Find Them](https://mlsanity.com/paper/2602.09128/) (2026)
- [Knowledge Distillation-Based Model Extraction Attack using GAN-based Private Counterfactual Explanations](https://mlsanity.com/paper/2404.03348/) (2024)
- [Causality-Aware Local Interpretable Model-Agnostic Explanations](https://mlsanity.com/paper/2212.05256/) (2022)
- [Counterfactual optimization for fault prevention in complex wind energy systems](https://mlsanity.com/paper/2507.08849/) (2025)
- [Density-Guided Robust Counterfactual Explanations on Tabular Data under Model Multiplicity](https://mlsanity.com/paper/2605.30901/) (2026)

## Abstract (arXiv)

Post-hoc explanation methods for machine learning models have been widely used to support decision-making. One of the popular methods is Counterfactual Explanation (CE), also known as Actionable Recourse, which provides a user with a perturbation vector of features that alters the prediction result. Given a perturbation vector, a user can interpret it as an "action" for obtaining one's desired decision result. In practice, however, showing only a perturbation vector is often insufficient for users to execute the action. The reason is that if there is an asymmetric interaction among features, such as causality, the total cost of the action is expected to depend on the order of changing features. Therefore, practical CE methods are required to provide an appropriate order of changing features in addition to a perturbation vector. For this purpose, we propose a new framework called Ordered Counterfactual Explanation (OrdCE). We introduce a new objective function that evaluates a pair of an action and an order based on feature interaction. To extract an optimal pair, we propose a mixed-integer linear optimization approach with our objective function. Numerical experiments on real datasets demonstrated the effectiveness of our OrdCE in comparison with unordered CE methods.