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Query Complexity of Search Problems

Authors Arkadev Chattopadhyay , Yogesh Dahiya , Meena Mahajan

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Arkadev Chattopadhyay
  • Tata Institute of Fundamental Research, Mumbai, India
Yogesh Dahiya
  • The Institute of Mathematical Sciences (a CI of Homi Bhabha National Institute), Chennai, India
Meena Mahajan
  • The Institute of Mathematical Sciences (a CI of Homi Bhabha National Institute), Chennai, India

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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)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Oracles and decision trees
  • Theory of computation → Models of computation
  • Decision trees
  • Search problems
  • Pseudo-determinism
  • Randomness


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