Easy Sets and Hard Certificate Schemes

TL;DR

This paper investigates the complexity of sets with easy certificates in NP, characterizing their properties, structural conditions, and limitations through complexity theory and relativization techniques.

Contribution

It provides new characterizations of sets with easy certificates using Kolmogorov complexity and establishes bounds and limitations via structural and relativization results.

Findings

Equivalent characterizations of classes with easy certificates

Structural conditions affecting class sizes and properties

Negative results on the limits of these classes using relativization

Abstract

Can easy sets only have easy certificate schemes? In this paper, we study the class of sets that, for all NP certificate schemes (i.e., NP machines), always have easy acceptance certificates (i.e., accepting paths) that can be computed in polynomial time. We also study the class of sets that, for all NP certificate schemes, infinitely often have easy acceptance certificates. In particular, we provide equivalent characterizations of these classes in terms of relative generalized Kolmogorov complexity, showing that they are robust. We also provide structural conditions---regarding immunity and class collapses---that put upper and lower bounds on the sizes of these two classes. Finally, we provide negative results showing that some of our positive claims are optimal with regard to being relativizable. Our negative results are proven using a novel observation: we show that the classical ``wide spacing'' oracle construction technique yields instant non-bi-immunity results. Furthermore, we establish a result that improves upon Baker, Gill, and Solovay's classical result that NP \neq P = NP \cap coNP holds in some relativized world.