Interior and exterior functions of positive Boolean functions

The interior and exterior functions of a Boolean function f were introduced in Makino and Ibaraki (Discrete Appl. Math. 69 (1996) 209-231), as stability (or robustness) measures of the f. In this paper, we investigate the complexity of two problems α-INTERIOR and α-EXTERIOR, introduced therein. We first answer the question about the complexity of α-INTERIOR left open in Makino and Ibaraki (Discrete Appl. Math. 69 (1996) 209-231); it has no polynomial total time algorithm even if α is bounded by a constant, unless P = NP. However, for positive h-term DNF functions with h bounded by a constant, problems α-INTERIOR and α-EXTERIOR can be solved in (input) polynomial time and polynomial delay, respectively. Furthermore, for positive k-DNF functions, α-INTERIOR for two cases in which k = 1, and α and k are both bounded by a constant, can be solved in polynomial delay and in polynomial time, respectively.