What is a hard instance of a computational problem?
In this paper a measure for the complexity of particular instances with respect to a given decision problem is introduced and investigated. Intuitively, an instance x is considered to be hard for a problem A if every algorithm that decides A and runs "fast" on x needs to look up (a description of) x in a table. A main result states that all problems not in P have infinitely many (polynomially) hard instances. Further, there exist problems in EXPTIME with all their instances being hard. The behavior of hard instances under polynomial reductions and the connections with complexity cores and circuits are studied.
KeywordsTuring Machine Complexity Core Kolmogorov Complexity Circuit Complexity Hard Instance
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