# Power indices and easier hard problems

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## Abstract

The concepts of power_index, satisfiability hypothesis (**SH**), and structure tree are introduced and used to make sharper hypotheses about a problem's complexity than “the problem is*NP*-complete.” These concepts are used to characterize the complexities of a number of basic*NP*-complete problems, including both CLIQUE and PARTITION which are shown to have power-indices at most 1/2. Also, the problem 3SAT is shown to be solvable deterministically in time exponential only in the**square root** of*v*+*c*, where*v* is the number of variables and*c* is the number of “crossovers” needed to layout the formula in the plane.

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## Copyright information

© Springer-Verlag New York Inc. 1990