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). Using our proof along with a decision-tree manipulation technique, we give a simple and self-contained proof that the \(\text{SearchCNF}\) problem on random \(\text{k}\) -CNF has pseudo-deterministic query complexity \(\Omega(n^{1/3})\) ; a lower bound of \(\Omega(\sqrt{n})\) is known, due to Goldwasser, Impagliazzo, Pitassi, and Santhanam (CCC'21), but via a significantly more complex proof.
We improve the known separation between pseudo-deterministic and randomized decision tree size for total search problems in two ways: (1) We exhibit an \(\text{exp}(\widetilde{\Omega}(n^{1/4}))\) separation for the \(\text{SearchCNF}\) relation for random \(k\) -CNFs. This seems to be the first exponential lower bound on the pseudo-deterministic size complexity of \(\text{SearchCNF}\) associated with random \(k\) -CNFs. (2) We exhibit an \({\text{exp}(\Omega(n))}\) separation for the \(\text{ApproxHW}\) relation. The previous best known separation for any relation was \({\text{exp}(\Omega(n^{1/2}))}\) .
We also separate pseudo-determinism from randomness in \(\text{AND}\) and \(\text{CONJ}\) decision trees, and determinism from pseudo-determinism in \(\text{Parity}\) decision trees.
Finally, for a hypercube colouring problem, that was introduced by Goldwasswer et al. to analyze the pseudo-deterministic complexity of a complete problem in \(\text{TFNPdt}\) , we prove that either the monotone block-sensitivity or the anti-monotone block sensitivity is \({\Omega(n^{1/3})}\) ; Goldwasser et al. showed an \({\Omega(n^{1/2})}\) bound for general block-sensitivity.
