The Complexity and Distribution of Hard Problems

• Authors: David W. Juedes;Jack H. Lutz
• Venue: SIAM Journal on Computing
• Year: 1995

Measure-theoretic aspects of the $\leq^{\rm P}_{\rm m}$-reducibility structure of the exponential time complexity classes E=DTIME($2^{\rm linear}$) and $E_{2}={\rm DTIME}(2^{\rm polynomial})$ are investigated. Particular attention is given to the complexity (measured by the size of complexity cores) and distribution (abundance in the sense of measure) of languages that are $\leq^{\rm P}_{\rm m}$-hard for E and other complexity classes. Tight upper and lower bounds on the size of complexity cores of hard languages are derived. The upper bound says that the $\leq^{\rm P}_{\rm m}$-hard languages for E are unusually simple, in the sense that they have smaller complexity cores than most languages in E. It follows that the $\leq^{\rm P}_{\rm m}$-complete languages for E form a measure 0 subset of E (and similarly in $E_2$). This latter fact is seen to be a special case of a more general theorem, namely, that {\it every} \pmr-degree (e.g., the degree of all \pmr-complete languages for NP) has measure 0 in E and in \Ep.