On W[1]-Hardness as Evidence for Intractability

The central conjecture of parameterized complexity states that FPT 6= W[1], and is generally regarded as the parameterized counterpart to P 6= NP. We revisit the issue of the plausibility of FPT 6= 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 has consequences for the A-Hierarchy as well, namely that unless W[P] is a subset of some class 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, downward separation holds for the A-Hierarchy. Finally, we give weak (oracle-based) evidence that the inclusion W[t] ⊆ 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).