Complete and incomplete randomized NP problems

Abstract

A randomized decision problem is a decision problem together with a probability function on the instances. Leonid Levin [Lev1] generalized the NP completeness theory to the case of properly defined randomized NP (shortly, RNP) problems and proved the completeness of a randomized version of the bounded tiling problem with respect to (appropriately generalized) Ptime reductions. Levin's proof naturally splits into two parts; a randomized version of the bounded halting problem is proved complete and then reduced to Randomized Tiling. David Johnson [Jo] provided some intuition behind Levin's definitions and proofs, and challenged readers to find additional natural complete RNP problems.