Counterfactual explanations (CEs) offer a human-understandable way to explain decisions by identifying specific changes to the input parameters of a base or present model that would lead to a desired change in the outcome. For optimization models, CEs have primarily been studied in limited contexts and little research has been done on CEs for general integer optimization problems. In this work, we address this gap. We first show that the general problem of constructing a CE is $\Sigma_2^p$-complete even for binary integer programs with just a single mutable constraint. Second, we propose solution algorithms for several of the most tractable special cases: (i) mutable objective parameters, (ii) a single mutable constraint, (iii) mutable right-hand-side, and (iv) all input parameters can be modified. We evaluate our approach using classical knapsack problem instances, focusing on cases with mutable constraint parameters. Our results show that our methods are capable of finding optimal CEs for small instances involving up to 40 items within a few hours.

翻译：反事实解释（CEs）通过识别对基础或现有模型输入参数的具体更改（这些更改将导致结果发生期望的变化），提供了一种人类可理解的方式来解释决策。对于优化模型，反事实解释主要在有限背景下被研究，针对一般整数优化问题的反事实解释研究甚少。本工作旨在填补这一空白。我们首先证明，即使对于仅具有单个可变约束的二元整数规划，构建反事实解释的一般问题也是$\Sigma_2^p$完全的。其次，我们针对几个最易处理的特殊情况提出了求解算法：（i）可变目标参数，（ii）单个可变约束，（iii）可变右端项，以及（iv）所有输入参数均可修改。我们使用经典背包问题实例评估了我们的方法，重点关注具有可变约束参数的情况。结果表明，我们的方法能够在数小时内为涉及多达40个项目的小规模实例找到最优反事实解释。