Integer Linear Programming (ILP) has a broad range of applications in various areas of artificial intelligence. Yet in spite of recent advances, we still lack a thorough understanding of which structural restrictions make ILP tractable. Here we study ILP instances consisting of a small number of “global” variables and/or constraints such that the remaining part of the instance consists of small and otherwise independent components; this is captured in terms of a structural measure we call fracture backdoors which generalizes, for instance, the well-studied class of N-fold ILP instances.

Our main contributions can be divided into three parts. First, we formally develop fracture backdoors and obtain exact and approximation algorithms for computing these. Second, we exploit these backdoors to develop several new parameterized algorithms for ILP; the performance of these algorithms will naturally scale based on the number of global variables or constraints in the instance. Finally, we complement the developed algorithms with matching lower bounds. Altogether, our results paint a near-complete complexity landscape of ILP with respect to fracture backdoors.¹

整数线性规划 (ILP) 在人工智能的各个领域都有广泛的应用。然而，尽管最近取得了进展，我们仍然缺乏对哪些结构限制使 ILP 易于处理的透彻了解。在这里，我们研究由少量“全局”变量和/或约束组成的 ILP 实例，以便实例的其余部分由小的和其他独立的组件组成；这是根据我们称为断裂后门的结构度量来捕获的，例如，它概括了经过充分研究的N折 ILP 实例类别。

我们的主要贡献可以分为三个部分。首先，我们正式开发了裂缝后门，并获得了计算这些后门的精确和近似算法。其次，我们利用这些后门为 ILP 开发了几种新的参数化算法；这些算法的性能自然会根据实例中全局变量或约束的数量进行扩展。最后，我们用匹配的下界补充开发的算法。总而言之，我们的结果描绘了关于断裂后门的 ILP 近乎完整的复杂性图景。¹