Computer-Aided complexity classification of combinational problems

We describe a computer program that has been used to maintain a record of the known complexity results for a class of 4536 machine scheduling problems. The input of the program consists of a listing of known “easy” problems and a listing of known “hard” problems. The program employs the structure of the problem class to determine the implications of these results. The output provides a listing of essential results in the form of maximal easy and minimal hard problems as well as listings of minimal and maximal open problems, which are helpful in indicating the direction of future research. The application of the program to a restricted class of 120 single-machine problems is demonstrated. Possible refinements and extensions to other research areas are suggested. It is also shown that the problem of determining the minimum number of results needed to resolve the status of all remaining open problems in a complexity classification such as ours is itself a hard problem.

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