On Optimal Algorithms and Optimal Proof Systems

Abstract

A deterministic algorithm O accepting a language L is called (polynomially) optimal if for any algorithm A accepting L there is a polynomial p such that timeO(x) ≤ p(|x|+ timeA(x)) for every x ∈ L. It is shown that an optimal acceptor for a language L exists if there is a p-optimal proof system for L. If L is a p-cylinder also the inverse implication holds. This result widely generalizes work from Krajicek and Pudlak who showed the result for L = TAUT. It is further shown how to construct an optimal acceptor for a p-cylinder L, given an acceptor for L which runs fast on every easy subset of L. Then we investigate the relationship of this notion of an 'optimal acceptor' to a more general notion of optimality. Here, instead of considering time-complexity on each individual string x, worst-case time-bounds are considered. It is observed that every set complete for exponential time under linearly length-bounded polynomial-time many-one reducibility has an acceptor with an optimal time-bound whereas on the other hand no set hard for exponential time under polynomial-time many-one reducibility has a p-optimal proof system. Finally we show how these results can be translated to nondeterministic algorithms and optimal proof systems.