Recently, Brand, Ganian and Simonov introduced a parameterized refinement of the classical PAC-learning sample complexity framework. A crucial outcome of their investigation is that for a very wide range of learning problems, there is a direct and provable correspondence between fixed-parameter PAC-learnability (in the sample complexity setting) and the fixed-parameter tractability of a corresponding "consistency checking" search problem (in the setting of computational complexity). The latter can be seen as generalizations of classical search problems where instead of receiving a single instance, one receives multiple yes- and no-examples and is tasked with finding a solution which is consistent with the provided examples. Apart from a few initial results, consistency checking problems are almost entirely unexplored from a parameterized complexity perspective. In this article, we provide an overview of these problems and their connection to parameterized sample complexity, with the primary aim of facilitating further research in this direction. Afterwards, we establish the fixed-parameter (in)-tractability for some of the arguably most natural consistency checking problems on graphs, and show that their complexity-theoretic behavior is surprisingly very different from that of classical decision problems. Our new results cover consistency checking variants of problems as diverse as (k-)Path, Matching, 2-Coloring, Independent Set and Dominating Set, among others.

翻译：最近，Brand、Ganian和Simonov引入了经典PAC学习样本复杂性框架的参数化精炼。其研究的关键成果在于，对于范围极广的学习问题，固定参数PAC可学习性（在样本复杂性设定中）与相应“一致性检测”搜索问题的固定参数可处理性（在计算复杂性设定中）之间存在直接且可证明的对应关系。后者可视为经典搜索问题的推广——不再接收单个实例，而是接收多个正例和反例，任务在于找出与所提供样例一致的一个解。除少数初步结果外，一致性检测问题在参数化复杂性视角下几乎尚未被探索。本文概述了这些问题及其与参数化样本复杂性的关联，主要目标是推动该方向的进一步研究。随后，我们确立了图论中若干公认最自然的一致性检测问题的固定参数（不）可处理性，并显示其复杂性理论行为与经典判定问题相比出人意料地迥异。我们的新结果覆盖了诸如（k-）路径、匹配、二染色、独立集和支配集等多样问题的一致性检测变体。