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 Î 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 Krajíčcek and Pudlák 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.
CITATION STYLE
Messner, J. (1999). On optimal algorithms and optimal proof systems. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 1563, pp. 541–550). Springer Verlag. https://doi.org/10.1007/3-540-49116-3_51
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