On W[1]-Hardness as Evidence for Intractability

TL;DR

This paper investigates the complexity-theoretic foundations of the W[1]-hardness class, providing evidence that separating FPT from W[1] is significantly more difficult than the P versus NP problem, and explores implications for hierarchies in parameterized complexity.

Contribution

It offers new theoretical evidence on the hardness of separating FPT from W[1], and analyzes the structural relationships between various parameterized complexity classes and hierarchies.

Findings

W[1] closure under relativization with FPT-oracles equals W[P]

Structural differences between the A-Hierarchy and Polynomial Hierarchy

W[t] is strictly contained in A[t] for t>1

Abstract

The central conjecture of parameterized complexity states that FPT is not equal to W[1], and is generally regarded as the parameterized counterpart to P != NP. We revisit the issue of the plausibility of FPT != W[1], focusing on two aspects: the difficulty of proving the conjecture (assuming it holds), and how the relation between the two classes might differ from the one between P and NP. Regarding the first aspect, we give new evidence that separating FPT from W[1] would be considerably harder than doing the same for P and NP. Our main result regarding the relation between FPT and W[1] states that the closure of W[1] under relativization with FPT-oracles is precisely the class W[P], implying that either FPT is not low for W[1], or the W-Hierarchy collapses. This theorem also has consequences for the A-Hierarchy (a parameterized version of the Polynomial Hierarchy), namely that unless W[P] is a subset of some level A[t], there are structural differences between the A-Hierarchy and the Polynomial Hierarchy. We also prove that under the unlikely assumption that W[P] collapses to W[1] in a specific way, the collapse of any two consecutive levels of the A-Hierarchy implies the collapse of the entire hierarchy to a finite level; this extends a result of Chen, Flum, and Grohe (2005). Finally, we give weak (oracle-based) evidence that the inclusion of W[t] in A[t] is strict for t>1, and that the W-Hierarchy is proper. The latter result answers a question of Downey and Fellows (1993).