On W-Hardness as Evidence for Intractability
The central conjecture of parameterized complexity states that FPT !=W, and is generally regarded as the parameterized counterpart to P !=NP. We revisit the issue of the plausibility of FPT !=W, 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 would be considerably harder than doing the same for P and NP. Our main result regarding the relation between FPT and W states that the closure of W under relativization with FPT-oracles is precisely the class W[P], implying that either FPT is not low for W, 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 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 W[t]subseteqA[t] is strict for t>1, and that the W-Hierarchy is proper. The latter result answers a question of Downey and Fellows (1993).
|Leibniz International Proceedings in Informatics|
|43rd International Symposium on Mathematical Foundations of Computer Science|
|Organisation||Algorithms and Complexity|
Bottesch, R.C. (2018). On W-Hardness as Evidence for Intractability. In Leibniz International Proceedings in Informatics. doi:10.4230/LIPIcs.MFCS.2018.73