2 Search Results for "Dahiya, Yogesh"


Document
Query Complexity of Search Problems

Authors: Arkadev Chattopadhyay, Yogesh Dahiya, and Meena Mahajan

Published in: LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)


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.

Cite as

Arkadev Chattopadhyay, Yogesh Dahiya, and Meena Mahajan. Query Complexity of Search Problems. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 34:1-34:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@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/entities/document/10.4230/LIPIcs.MFCS.2023.34},
  URN =		{urn:nbn:de:0030-drops-185689},
  doi =		{10.4230/LIPIcs.MFCS.2023.34},
  annote =	{Keywords: Decision trees, Search problems, Pseudo-determinism, Randomness}
}
Document
On (Simple) Decision Tree Rank

Authors: Yogesh Dahiya and Meena Mahajan

Published in: LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)


Abstract
In the decision tree computation model for Boolean functions, the depth corresponds to query complexity, and size corresponds to storage space. The depth measure is the most well-studied one, and is known to be polynomially related to several non-computational complexity measures of functions such as certificate complexity. The size measure is also studied, but to a lesser extent. Another decision tree measure that has received very little attention is the minimal rank of the decision tree, first introduced by Ehrenfeucht and Haussler in 1989. This measure is not polynomially related to depth, and hence it can reveal additional information about the complexity of a function. It is characterised by the value of a Prover-Delayer game first proposed by Pudlák and Impagliazzo in the context of tree-like resolution proofs. In this paper we study this measure further. We obtain upper and lower bounds on rank in terms of (variants of) certificate complexity. We also obtain upper and lower bounds on the rank for composed functions in terms of the depth of the outer function and the rank of the inner function. We compute the rank exactly for several natural functions and use them to show that all the bounds we have obtained are tight. We also observe that the size-rank relationship for decision trees, obtained by Ehrenfeucht and Haussler, is tight upto constant factors.

Cite as

Yogesh Dahiya and Meena Mahajan. On (Simple) Decision Tree Rank. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 15:1-15:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


Copy BibTex To Clipboard

@InProceedings{dahiya_et_al:LIPIcs.FSTTCS.2021.15,
  author =	{Dahiya, Yogesh and Mahajan, Meena},
  title =	{{On (Simple) Decision Tree Rank}},
  booktitle =	{41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)},
  pages =	{15:1--15:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-215-0},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{213},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.15},
  URN =		{urn:nbn:de:0030-drops-155263},
  doi =		{10.4230/LIPIcs.FSTTCS.2021.15},
  annote =	{Keywords: Boolean functions, Decision trees, certificate complexity, rank}
}
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