On the Complexity and Inapproximability of Shortest Implicant Problems

We investigate the complexity and approximability of a basic optimization problem in the second level of the Polynomial Hierarchy, that of finding shortest implicants. We show that the DNF variant of this problem is complete for a complexity class in the second level of the hierarchy utilizing log2 n-limited nondeterminism. We obtain inapproximability results for the DNF and formula variants of the shortest implicant problem that show that trivial approximation algorithms are optimal for these problems, up to lower order terms. It is hoped that these results will be useful in studying the complexity and approximability of circuit minimization problems, which have close connections to implicant problems.

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