Hard Instances of Hard Problems

This paper investigates the instance complexities of problems that are hard or weakly hard for exponential time under polynomial time many one reductions It is shown that almost every instance of almost every problem in exponential time has essentially maximal instance complexity It follows that every weakly hard problem has a dense set of such maximally hard instances This extends the theorem due to Orponen Ko Sch oning and Watanabe that every hard prob lem for exponential time has a dense set of maximally hard instances Complementing this it is shown that every hard problem for exponen tial time also has a dense set of unusually easy instances