License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2023.34
URN: urn:nbn:de:0030-drops-185689
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/18568/
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Chattopadhyay, Arkadev ; Dahiya, Yogesh ; Mahajan, Meena

Query Complexity of Search Problems

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Abstract

We relate various complexity measures like sensitivity, block sensitivity, certificate complexity for multi-output functions to the query complexities of such functions. Using these relations, we provide the following improvements upon the known relationship between pseudo-deterministic and deterministic query complexity for total search problems:
- We show that deterministic query complexity is at most the third power of its pseudo-deterministic query complexity. Previously, a fourth-power relation was shown by Goldreich, Goldwasser and Ron (ITCS'13).
- We improve the known separation between pseudo-deterministic and randomized decision tree size for total search problems in two ways: (1) we exhibit an exp(Ω̃(n^{1/4})) separation for the SearchCNF relation for random k-CNFs. This seems to be the first exponential lower bound on the pseudo-deterministic size complexity of SearchCNF associated with random k-CNFs. (2) we exhibit an exp(Ω(n)) separation for the ApproxHamWt relation. The previous best known separation for any relation was exp(Ω(n^{1/2})). We also separate pseudo-determinism from randomness in And and (And,Or) decision trees, and determinism from pseudo-determinism in Parity decision trees. For a hypercube colouring problem, that was introduced by Goldwasswer, Impagliazzo, Pitassi and Santhanam (CCC'21) to analyze the pseudo-deterministic complexity of a complete problem in TFNP^{dt}, we prove that either the monotone block-sensitivity or the anti-monotone block sensitivity is Ω(n^{1/3}); Goldwasser et al. showed an Ω(n^{1/2}) bound for general block-sensitivity.

BibTeX - Entry

@InProceedings{chattopadhyay_et_al:LIPIcs.MFCS.2023.34,
  author =	{Chattopadhyay, Arkadev and Dahiya, Yogesh and Mahajan, Meena},
  title =	{{Query Complexity of Search Problems}},
  booktitle =	{48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
  pages =	{34:1--34:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-292-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{272},
  editor =	{Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/18568},
  URN =		{urn:nbn:de:0030-drops-185689},
  doi =		{10.4230/LIPIcs.MFCS.2023.34},
  annote =	{Keywords: Decision trees, Search problems, Pseudo-determinism, Randomness}
}

Keywords: Decision trees, Search problems, Pseudo-determinism, Randomness
Collection: 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)
Issue Date: 2023
Date of publication: 21.08.2023

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