State minimization of combinatorial filters is a fundamental problem that arises, for example, in building cheap, resource-efficient robots. But exact minimization is known to be NP-hard. This paper conducts a more nuanced analysis of this hardness than up till now, and uncovers two factors which contribute to this complexity. We show each factor is a distinct source of the problem's hardness and are able, thereby, to shed some light on the role played by (1) structure of the graph that encodes compatibility relationships, and (2) determinism-enforcing constraints. Just as a line of prior work has sought to introduce additional assumptions and identify sub-classes that lead to practical state reduction, we next use this new, sharper understanding to explore special cases for which exact minimization is efficient. We introduce a new algorithm for constraint repair that applies to a large sub-class of filters, subsuming three distinct special cases for which the possibility of optimal minimization in polynomial time was known earlier. While the efficiency in each of these three cases previously appeared to stem from seemingly dissimilar properties, when seen through the lens of the present work, their commonality now becomes clear. We also provide entirely new families of filters that are efficiently reducible.

翻译：组合滤波器的状态最小化是一个基础性问题，例如在构建低成本、资源高效的机器人时至关重要。然而，精确最小化已知是NP难题。本文对这一问题复杂度进行了比以往更精细的分析，揭示了导致该复杂度的两个因素。我们证明每个因素都是问题难度的独立来源，从而阐明了（1）编码兼容性关系的图结构，以及（2）确定性强制约束各自所起的作用。正如先前一系列研究试图引入额外假设并识别出能实现实用状态缩减的子类，我们接下来利用这一新的更清晰认知，探索可实现高效精确最小化的特例。我们提出了一种约束修复新算法，适用于一大类滤波器，涵盖了先前已知可在多项式时间内实现最优最小化的三个不同特例。尽管这三个案例的高效性看似源于迥异的性质，但通过本文视角审视，其共性变得清晰。此外，我们还提供了全新的可高效约简的滤波器族系。