On the Structure and Complexity of Infinite Sets with Minimal Perfect Hash Functions

This paper studies the class of infinite sets that have minimal perfect hash functions - one-to-one onto maps between the sets and Σ*-computable in polynomial time. We show that all standard NP-complete sets have polynomial-time computable minimal perfect hash functions, and give a structural condition sufficient to ensure that all infinite NP sets have polynomial-time computable minimal perfect hash functions. If E = Σ E/2, then all infinite NP sets have polynomial-time computable minimal perfect hash functions. On the other hand, we present evidence that some infinite NP sets, and indeed some infinite P sets, do not have polynomial-time computable minimal perfect hash functions: if an infinite NP set A has polynomial-time computable perfect minimal hash functions, then A has an infinite sparse NP subset, yet we construct a relativized world in which some infinite NP sets lack infinite sparse NP subsets. This world is built upon a result that is of interest in its own right; we determine optimally -- with respect to any relativizable proof technique -- the complexity of the easiest infinite sparse subsets that infinite P sets are guaranteed to have