Algorithmic Meta-Theorems

TL;DR

This paper surveys recent developments in algorithmic meta-theorems, highlighting their logical and structural components, and discusses the graph minor theory underpinning many of these results.

Contribution

It provides a comprehensive overview of recent meta-theorems and the proof techniques, especially those related to graph minor theory, used in their establishment.

Findings

Summarizes key algorithmic meta-theorems and their applications.

Explains the role of graph minor theory in proving meta-theorems.

Highlights the logical and structural components of meta-theorems.

Abstract

Algorithmic meta-theorems are general algorithmic results applying to a whole range of problems, rather than just to a single problem alone. They often have a "logical" and a "structural" component, that is they are results of the form: every computational problem that can be formalised in a given logic L can be solved efficiently on every class C of structures satisfying certain conditions. This paper gives a survey of algorithmic meta-theorems obtained in recent years and the methods used to prove them. As many meta-theorems use results from graph minor theory, we give a brief introduction to the theory developed by Robertson and Seymour for their proof of the graph minor theorem and state the main algorithmic consequences of this theory as far as they are needed in the theory of algorithmic meta-theorems.