Trustworthiness in artificial intelligence depends not only on what a model decides, but also on how it handles and explains cases in which a reliable decision cannot be made. In critical domains such as healthcare and finance, a reject option allows the model to abstain when evidence is insufficient, making it essential to explain why an instance is rejected in order to support informed human intervention. In these settings, explanations must not only be interpretable, but also faithful to the underlying model and computationally efficient enough to support real-time decision making. Abductive explanations guarantee fidelity, but their exact computation is known to be NP-hard for many classes of models, limiting their practical applicability. Computing \textbf{minimum-size} abductive explanations is an even more challenging problem, as it requires reasoning not only about fidelity but also about optimality. Prior work has addressed this challenge in restricted settings, including log-linear-time algorithms for computing minimum-size abductive explanations in linear models without rejection, as well as a polynomial-time method based on linear programming for computing abductive explanations, without guarantees of minimum size, for linear models with a reject option. In this work, we bridge these lines of research by computing minimum-size abductive explanations for linear models with a reject option. For accepted instances, we adapt the log-linear algorithm to efficiently compute optimal explanations. For rejected instances, we formulate a 0-1 integer linear programming problem that characterizes minimum-size abductive explanations of rejection. Although this formulation is NP-hard in theory, our experimental results show that it is consistently more efficient in practice than the linear-programming-based approach that does not guarantee minimum-size explanations.

翻译：人工智能的可信度不仅取决于模型做出的决策，还取决于其如何处理和解释无法做出可靠决策的情况。在医疗和金融等关键领域，拒绝选项允许模型在证据不足时选择弃权，因此解释实例被拒绝的原因对于支持知情的人工干预至关重要。在这些场景中，解释不仅需要可解释，还必须忠实于底层模型，并且计算效率足够高以支持实时决策。溯因解释能保证忠实性，但其精确计算已知对许多模型类别是NP难的，这限制了其实际应用。计算**最小规模**的溯因解释是一个更具挑战性的问题，因为它不仅需要考虑忠实性，还需要考虑最优性。先前的研究已在受限场景中应对了这一挑战，包括为无拒绝选项的线性模型计算最小规模溯因解释的对数线性时间算法，以及为带拒绝选项的线性模型计算溯因解释（不保证最小规模）的基于线性规划的多项式时间方法。在本工作中，我们通过为带拒绝选项的线性模型计算最小规模溯因解释，弥合了这些研究方向。对于被接受的实例，我们调整了对数线性算法以高效计算最优解释。对于被拒绝的实例，我们构建了一个0-1整数线性规划问题来刻画拒绝的最小规模溯因解释。尽管该公式在理论上是NP难的，但我们的实验结果表明，在实践中它始终比不保证解释最小规模的基于线性规划的方法更高效。