The complexity of central slice functions

Abstract

Let f be a monotone Boolean function over X = {x₁,…,x_n}. The k-slice of f is the function f_k = (f^T_kⁿ)vT_k+1ⁿ, where T_kⁿ is the kth threshold function.

Berkowitz has shown that sufficiently large superlinear lower bounds on the monotone network complexity of k_k imply lower bounds of the same order on the combinational complexity of f, and that if f has large combinational complexity, then some slice of f must have large monotone complexity. However, this latter result does not specify any particular slice and it is known that some (nontrivial) slice functions of NP-complete predicates have linear complexity. In this paper we consider the $\text{1}\text{2}\text{n}$ slice functions and show that, for three basic monotone Boolean NP-complete functions, this slice is also NP-complete.

In addition the $\text{1}\text{2}\text{n-}\text{slice}$ of some Boolean matrix functions F is studied, and is proved to be no easier to compute than F.