A ZPP\u3csup\u3eNP[1]\u3c/sup\u3e lifting theorem

The complexity class ZPPNP[1] (corresponding to zero-error randomized algorithms with access to one NP oracle query) is known to have a number of curious properties. We further explore this class in the settings of time complexity, query complexity, and communication complexity. For starters, we provide a new characterization: ZPPNP[1] equals the restriction of BPPNP[1] where the algorithm is only allowed to err when it forgoes the opportunity to make an NP oracle query. Using the above characterization, we prove a query-to-communication lifting theorem, which translates any ZPPNP[1] decision tree lower bound for a function f into a ZPPNP[1] communication lower bound for a two-party version of f. As an application, we use the above lifting theorem to prove that the ZPPNP[1] communication lower bound technique introduced by Göös, Pitassi, and Watson (ICALP 2016) is not tight. We also provide a “primal” characterization of this lower bound technique as a complexity class