Polynomial kernels collapse the W-hierarchy

TL;DR

This paper demonstrates that the existence of polynomial kernels for many FPT problems implies the collapse of the W-hierarchy, revealing a deep connection between kernelizability and parameterized intractability.

Contribution

It establishes the first link between polynomial kernelizability and the collapse of the W-hierarchy in parameterized complexity theory.

Findings

Polynomial kernels imply W[P] = FPT.

Exponential kernels relate to sub-exponential algorithms.

Kernelizability impacts the structure of the FPT class.

Abstract

We prove that, for many parameterized problems in the class FPT, the existence of polynomial kernels implies the collapse of the W-hierarchy (i.e., W[P] = FPT). The collapsing results are also extended to assumed exponential kernels for problems in the class FPT. In particular, we establish a close relationship between polynomial (and exponential) kernelizability and the existence of sub-exponential time algorithms for a spectrum of circuit satisfiability problems in FPT. To the best of our knowledge, this is the first work that connects hardness for polynomial kernelizability of FPT problems to parameterized intractability. Our work also offers some new insights into the class FPT.