Inapproximability of a Pair of Forms Defining a Partial Boolean Function

TL;DR

This paper proves that jointly minimizing two Boolean forms under certain constraints is computationally hard, even harder than the single-function case, with no efficient approximation algorithms available under standard complexity assumptions.

Contribution

It demonstrates the inapproximability of jointly minimizing pairs of Boolean forms like decision trees and DNF, refuting the hypothesis that this problem is easier than single-function minimization.

Findings

Joint minimization is as hard as or harder than single-function minimization.

No polynomial-time approximation within a factor of o(log(|A|+|B|-1)) unless P=NP.

Disjunctive normal forms are at least as hard as MIN-SET-COVER.

Abstract

We consider the problem of jointly minimizing forms of two Boolean functions $f, g \colon \{0,1\}^J \to \{0,1\}$ such that $f + g \leq 1$ and so as to separate disjoint sets $A \cup B \subseteq \{0,1\}^J$ such that $f(A) = \{1\}$ and $g(B) = \{1\}$. We hypothesize that this problem is easier to solve or approximate than the well-understood problem of minimizing the form of one Boolean function $h: \{0,1\}^J \to \{0,1\}$ such that $h(A) = \{1\}$ and $h(B) = \{0\}$. For a large class of forms, including binary decision trees and ordered binary decision diagrams, we refute this hypothesis. For disjunctive normal forms, we show that the problem is at least as hard as MIN-SET-COVER. For all these forms, we establish that no $o(\ln (|A| + |B| -1))$-approximation algorithm exists unless P$=$NP.