Abstract
Algorithmic metatheorems state that if a problem can be described in a certain logic and the inputs are structured in a certain way, then the problem can be solved with a certain amount of resources. As an example, by Courcelle’s Theorem, all monadic second-order (“in a certain logic”) properties of graphs of bounded tree width (“structured in a certain way”) can be solved in linear time (“with a certain amount of resources”). Such theorems have become valuable tools in algorithmics: if a problem happens to have the right structure and can be described in the right logic, they immediately yield a (typically tight) upper bound on the time complexity of the problem. Perhaps even more importantly, several complex algorithms rely on algorithmic metatheorems internally to solve subproblems, which considerably broadens the range of applications of these theorems. This paper is intended as a gentle introduction to the ideas behind algorithmic metatheorems, especially behind some recent results concerning space and circuit classes, and tries to give a flavor of the range of their applications.
Highlights
How would a first-year student of computer science approach the problem of choosing at most k vertices from a graph such that all edges have at least one of their endpoints chosen? Most readers will know, that this is an NP-complete vertex cover problem, and you are most likely mentally weighing the different tools at your disposal for attacking such problems from the vast machinery developed in complexity theory
Algorithmic metatheorems tell us when and why the divide-and-conquer approach works for certain problems
They are statements following the pattern “If a problem can be described in a certain logic and can be decomposed in a certain way, the problem can be solved with a certain amount of resources”
Summary
How would a first-year student of computer science approach the problem of choosing at most k vertices from a graph such that all edges have at least one of their endpoints chosen? Most readers will know, that this is an NP-complete vertex cover problem, and you are most likely mentally weighing the different tools at your disposal for attacking such problems from the vast machinery developed in complexity theory. How would a first-year student of computer science approach the problem of choosing at most k vertices from a graph such that all edges have at least one of their endpoints chosen? She would first try to apply the arguably most important and ubiquitous algorithmic approach in computer science: divide-and-conquer. After all, she has seen that this approach lies at the heart of fundamental algorithms in computer science (like merge-sort, quick-sort, or binary search), and she has been told that it is routinely used in advanced algorithms (such as the “search trees of fixed-parameter algorithms”, whatever these may be, she wonders)
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