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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2020.9

The Power of Many Samples in Query Complexity

Authors: Andrew Bassilakis, Andrew Drucker, Mika Göös, Lunjia Hu, Weiyun Ma, and Li-Yang Tan

Published in: LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Abstract

The randomized query complexity 𝖱(f) of a boolean function f: {0,1}ⁿ → {0,1} is famously characterized (via Yao’s minimax) by the least number of queries needed to distinguish a distribution 𝒟₀ over 0-inputs from a distribution 𝒟₁ over 1-inputs, maximized over all pairs (𝒟₀,𝒟₁). We ask: Does this task become easier if we allow query access to infinitely many samples from either 𝒟₀ or 𝒟₁? We show the answer is no: There exists a hard pair (𝒟₀,𝒟₁) such that distinguishing 𝒟₀^∞ from 𝒟₁^∞ requires Θ(𝖱(f)) many queries. As an application, we show that for any composed function f∘g we have 𝖱(f∘g) ≥ Ω(fbs(f)𝖱(g)) where fbs denotes fractional block sensitivity.

Cite as

Andrew Bassilakis, Andrew Drucker, Mika Göös, Lunjia Hu, Weiyun Ma, and Li-Yang Tan. The Power of Many Samples in Query Complexity. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 9:1-9:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bassilakis_et_al:LIPIcs.ICALP.2020.9,
  author =	{Bassilakis, Andrew and Drucker, Andrew and G\"{o}\"{o}s, Mika and Hu, Lunjia and Ma, Weiyun and Tan, Li-Yang},
  title =	{{The Power of Many Samples in Query Complexity}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{9:1--9:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.9},
  URN =		{urn:nbn:de:0030-drops-124163},
  doi =		{10.4230/LIPIcs.ICALP.2020.9},
  annote =	{Keywords: Query complexity, Composition theorems}
}

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DOI: 10.4230/LIPIcs.ESA.2016.36

Exponential Time Paradigms Through the Polynomial Time Lens

Authors: Andrew Drucker, Jesper Nederlof, and Rahul Santhanam

Published in: LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

Abstract

We propose a general approach to modelling algorithmic paradigms for the exact solution of NP-hard problems. Our approach is based on polynomial time reductions to succinct versions of problems solvable in polynomial time. We use this viewpoint to explore and compare the power of paradigms such as branching and dynamic programming, and to shed light on the true complexity of various problems. As one instantiation, we model branching using the notion of witness compression, i.e., reducibility to the circuit satisfiability problem parameterized by the number of variables of the circuit. We show this is equivalent to the previously studied notion of `OPP-algorithms', and provide a technique for proving conditional lower bounds for witness compressions via a constructive variant of AND-composition, which is a notion previously studied in theory of preprocessing. In the context of parameterized complexity we use this to show that problems such as Pathwidth and Treewidth and Independent Set parameterized by pathwidth do not have witness compression, assuming NP subseteq coNP/poly. Since these problems admit fast fixed parameter tractable algorithms via dynamic programming, this shows that dynamic programming can be stronger than branching, under a standard complexity hypothesis. Our approach has applications outside parameterized complexity as well: for example, we show if a polynomial time algorithm outputs a maximum independent set of a given planar graph on n vertices with probability exp(-n^{1-epsilon}) for some epsilon>0, then NP subseteq coNP/poly. This negative result dims the prospects for one very natural approach to sub-exponential time algorithms for problems on planar graphs. As two other illustrations (more exploratory) of our approach, we model algorithms based on inclusion-exclusion or group algebras via the notion of "parity compression", and we model a subclass of dynamic programming algorithms with the notion of "disjunctive dynamic programming". These models give us a way to naturally classify various parameterized problems with FPT algorithms. In the case of the dynamic programming model, we show that Independent Set parameterized by pathwidth is complete for this model.

Cite as

Andrew Drucker, Jesper Nederlof, and Rahul Santhanam. Exponential Time Paradigms Through the Polynomial Time Lens. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 36:1-36:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{drucker_et_al:LIPIcs.ESA.2016.36,
  author =	{Drucker, Andrew and Nederlof, Jesper and Santhanam, Rahul},
  title =	{{Exponential Time Paradigms Through the Polynomial Time Lens}},
  booktitle =	{24th Annual European Symposium on Algorithms (ESA 2016)},
  pages =	{36:1--36:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-015-6},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{57},
  editor =	{Sankowski, Piotr and Zaroliagis, Christos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.36},
  URN =		{urn:nbn:de:0030-drops-63871},
  doi =		{10.4230/LIPIcs.ESA.2016.36},
  annote =	{Keywords: exponential time paradigms, branching, dynamic programming, lower bounds}
}

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Documents authored by Drucker, Andrew

The Power of Many Samples in Query Complexity

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Exponential Time Paradigms Through the Polynomial Time Lens

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