License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2018.73
URN: urn:nbn:de:0030-drops-96559
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Bottesch, Ralph Christian

On W[1]-Hardness as Evidence for Intractability

LIPIcs-MFCS-2018-73.pdf (0.5 MB)


The central conjecture of parameterized complexity states that FPT !=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 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).

BibTeX - Entry

  author =	{Ralph Christian Bottesch},
  title =	{{On W[1]-Hardness as Evidence for Intractability}},
  booktitle =	{43rd International Symposium on Mathematical Foundations  of Computer Science (MFCS 2018)},
  pages =	{73:1--73:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-086-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{117},
  editor =	{Igor Potapov and Paul Spirakis and James Worrell},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-96559},
  doi =		{10.4230/LIPIcs.MFCS.2018.73},
  annote =	{Keywords: Parameterized complexity, Relativization}

Keywords: Parameterized complexity, Relativization
Collection: 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)
Issue Date: 2018
Date of publication: 27.08.2018

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