2 Search Results for "Bottesch, Ralph Christian"

Document
DOI: 10.4230/LIPIcs.MFCS.2018.73
On W[1]-Hardness as Evidence for Intractability

Authors: Ralph Christian Bottesch

Published in: LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

Abstract
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).
Cite as

Ralph Christian Bottesch. On W[1]-Hardness as Evidence for Intractability. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 73:1-73:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bottesch:LIPIcs.MFCS.2018.73,
  author =	{Bottesch, Ralph Christian},
  title =	{{On W\lbrack1\rbrack-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 =	{Potapov, Igor and Spirakis, Paul and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.73},
  URN =		{urn:nbn:de:0030-drops-96559},
  doi =		{10.4230/LIPIcs.MFCS.2018.73},
  annote =	{Keywords: Parameterized complexity, Relativization}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2015.544
Correlation in Hard Distributions in Communication Complexity

Authors: Ralph Christian Bottesch, Dmitry Gavinsky, and Hartmut Klauck

Published in: LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)

Abstract
We study the effect that the amount of correlation in a bipartite distribution has on the communication complexity of a problem under that distribution. We introduce a new family of complexity measures that interpolates between the two previously studied extreme cases: the (standard) randomised communication complexity and the case of distributional complexity under product distributions. - We give a tight characterisation of the randomised complexity of Disjointness under distributions with mutual information k, showing that it is Theta(sqrt(n(k+1))) for all 0 <= k <= n. This smoothly interpolates between the lower bounds of Babai, Frankl and Simon for the product distribution case (k=0), and the bound of Razborov for the randomised case. The upper bounds improve and generalise what was known for product distributions, and imply that any tight bound for Disjointness needs Omega(n) bits of mutual information in the corresponding distribution. - We study the same question in the distributional quantum setting, and show a lower bound of Omega((n(k+1))^{1/4}), and an upper bound (via constructing communication protocols), matching up to a logarithmic factor. - We show that there are total Boolean functions f_d that have distributional communication complexity O(log(n)) under all distributions of information up to o(n), while the (interactive) distributional complexity maximised over all distributions is Theta(log(d)) for n <= d <= 2^{n/100}. This shows, in particular, that the correlation needed to show that a problem is hard can be much larger than the communication complexity of the problem. - We show that in the setting of one-way communication under product distributions, the dependence of communication cost on the allowed error epsilon is multiplicative in log(1/epsilon) - the previous upper bounds had the dependence of more than 1/epsilon. This result, for the first time, explains how one-way communication complexity under product distributions is stronger than PAC-learning: both tasks are characterised by the VC-dimension, but have very different error dependence (learning from examples, it costs more to reduce the error).
Cite as

Ralph Christian Bottesch, Dmitry Gavinsky, and Hartmut Klauck. Correlation in Hard Distributions in Communication Complexity. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 544-572, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bottesch_et_al:LIPIcs.APPROX-RANDOM.2015.544,
  author =	{Bottesch, Ralph Christian and Gavinsky, Dmitry and Klauck, Hartmut},
  title =	{{Correlation in Hard Distributions in Communication Complexity}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{544--572},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-89-7},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{40},
  editor =	{Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.544},
  URN =		{urn:nbn:de:0030-drops-53234},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.544},
  annote =	{Keywords: communication complexity; information theory}
}
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