The Complexity and Distribution of Hard Problems

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1992-08-13
Authors
Juedes, David
Lutz, Jack
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Computer Science

Computer Science—the theory, representation, processing, communication and use of information—is fundamentally transforming every aspect of human endeavor. The Department of Computer Science at Iowa State University advances computational and information sciences through; 1. educational and research programs within and beyond the university; 2. active engagement to help define national and international research, and 3. educational agendas, and sustained commitment to graduating leaders for academia, industry and government.

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The Computer Science Department was officially established in 1969, with Robert Stewart serving as the founding Department Chair. Faculty were composed of joint appointments with Mathematics, Statistics, and Electrical Engineering. In 1969, the building which now houses the Computer Science department, then simply called the Computer Science building, was completed. Later it was named Atanasoff Hall. Throughout the 1980s to present, the department expanded and developed its teaching and research agendas to cover many areas of computing.

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1969-present

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Abstract

Measure-theoretic aspects of the polynomial-time many-one reducibility structure of the exponential time complexity classes E=DTIME(2^linear) and E2=DTIME(2^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 polynomial-time many-one 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 bounds say that the polynomial-time many-one 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 polynomial-time many-one complete languages for E form a measure 0 subset of E (and similarly in E2). This latter fact is seen to be a special case of a more general theorem, namely, that every polynomial-time many-one degree (e.g., the degree of all polynomial-time many-one complete languages for NP) has measure 0 in E and in E2.

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