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.

Introduction

Algorithmic meta-theorems are general algorithmic results applying to a whole range of problems, rather than just to a single problem alone. In this paper we will concentrate on meta-theorems that have a logical and a structural component, that is on 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.

The first such theorem is Courcelle's well-known result [13] stating that every problem definable in monadic second-order logic can be solved efficiently on any class of graphs of bounded tree-width.