FPT is Characterized by Useful Obstruction Sets

TL;DR

This paper characterizes fixed-parameter tractability (FPT) through the lens of obstruction sets, linking kernelization, obstructions, and composition techniques to unify and deepen understanding of FPT problems.

Contribution

It provides a new characterization of FPT problems via finite, computable obstruction sets and explores their connection to kernelization and composition methods.

Findings

Polynomial kernels correspond to polynomial-size obstruction sets.

Exponential-size minor-minimal obstructions are key in OR-cross-compositions.

Cross-compositions can rule out polynomial-size obstructions for certain problems.

Abstract

Many graph problems were first shown to be fixed-parameter tractable using the results of Robertson and Seymour on graph minors. We show that the combination of finite, computable, obstruction sets and efficient order tests is not just one way of obtaining strongly uniform FPT algorithms, but that all of FPT may be captured in this way. Our new characterization of FPT has a strong connection to the theory of kernelization, as we prove that problems with polynomial kernels can be characterized by obstruction sets whose elements have polynomial size. Consequently we investigate the interplay between the sizes of problem kernels and the sizes of the elements of such obstruction sets, obtaining several examples of how results in one area yield new insights in the other. We show how exponential-size minor-minimal obstructions for pathwidth k form the crucial ingredient in a novel OR-cross-composition for k-Pathwidth, complementing the trivial AND-composition that is known for this problem. In the other direction, we show that OR-cross-compositions into a parameterized problem can be used to rule out the existence of efficiently generated quasi-orders on its instances that characterize the NO-instances by polynomial-size obstructions.