Boolean networks are dynamical models of disease development in which the activation levels of genes are represented by binary variables. Given a Boolean network, controls represent mutations or medical treatments that fix the activation levels of selected genes so that all states in every attractor (i.e., long-term recurrent states) satisfy a desired phenotype. Our goal is to enumerate all minimal controls, identifying critical gene subsets in disease development and therapy. This problem has an inherent bilevel integer programming structure and is computationally challenging. We propose an infeasibility-based Benders decomposition, a logic-based Benders framework for bilevel integer programs with multiple subproblems. In our application, each subproblem finds a forbidden attractor of a given length and yields a problem-specific feasibility cut. We also propose an auxiliary IP called subspace separation that finds a Boolean subspace that includes multiple forbidden attractors and thereby strengthens the cut. Numerical experiments show that the resulting algorithms are much more scalable than state-of-the-art methods and that subspace separation substantially improves performance.

翻译：布尔网络是疾病发展的动力学模型，其中基因激活水平由二元变量表示。给定一个布尔网络，控制操作代表通过突变或医疗手段固定选定基因的激活水平，使得每个吸引子（即长期递归状态）中的所有状态均满足期望表型。我们的目标是枚举所有最小控制操作，从而识别疾病发展和治疗中的关键基因子集。该问题具有固有的双层整数规划结构，计算难度较大。我们提出了一种基于不可行性的Benders分解方法——一种针对含多个子问题的双层整数规划的逻辑型Benders框架。在具体应用中，每个子问题用于寻找给定长度的禁止吸引子，并生成针对特定问题的可行性割。此外，我们还提出了一种名为“子空间分离”的辅助整数规划，通过寻找包含多个禁止吸引子的布尔子空间来强化割平面。数值实验表明，所提算法的可扩展性显著优于现有方法，且子空间分离技术能大幅提升算法性能。