Consistency-Checking Problems: A Gateway to Parameterized Sample Complexity

TL;DR

This paper explores the parameterized complexity of consistency checking problems linked to PAC-learning, revealing their fixed-parameter tractability or intractability and highlighting differences from classical decision problems.

Contribution

It provides an overview of consistency checking problems in parameterized complexity and establishes their fixed-parameter tractability or intractability for key graph problems.

Findings

Some consistency checking problems are fixed-parameter tractable.

Complexity behavior differs from classical decision problems.

The study connects sample complexity with parameterized computational complexity.

Abstract

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.