Query Complexity of Search Problems

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 show that the deterministic query complexity of total search problems 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). Furthermore, 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( e Ω( 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.