On W[1]-Hardness as Evidence for Intractability

Ralph C. Bottesch

43rd International Symposium on Mathematical Foundations of Computer Science, Leibniz International Proceedings in Informatics 117, pp. 73:1 – 73:15, 2018.


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).


  PDF |    doi:10.4230/LIPIcs.MFCS.2018.73  |  © Creative Commons License – CC BY


author = "Ralph Christian Bottesch",
title = "{On W[1]-Hardness as Evidence for Intractability}",
booktitle = "43rd International Symposium on Mathematical Foundations
of Computer Science",
editor = "Igor Potapov and Paul Spirakis and James Worrell",
series = "Leibniz International Proceedings in Informatics",
volume = 117,
pages = "73:1--73:15",
year = 2018,
doi = "10.4230/LIPIcs.MFCS.2018.73",
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