Hardness Results for Approximate Pure Horn CNF Formulae Minimization

TL;DR

This paper proves strong hardness of approximation results for minimizing clauses and literals in pure Horn CNF formulas, showing these problems are computationally intractable under common complexity assumptions.

Contribution

It establishes the first exponential hardness of approximation bounds for pure Horn CNF minimization problems, even under restricted input conditions.

Findings

No polynomial-time approximation within 2^{log^{1-o(1)} n} factor unless P=NP.

Even sub-exponential algorithms cannot achieve constant factor approximations unless ETH fails.

Hardness results hold for pure Horn 3-CNFs with O(n^{1+ε}) clauses.

Abstract

We study the hardness of approximation of clause minimum and literal minimum representations of pure Horn functions in $n$ Boolean variables. We show that unless P=NP, it is not possible to approximate in polynomial time the minimum number of clauses and the minimum number of literals of pure Horn CNF representations to within a factor of $2^{\log^{1-o(1)} n}$. This is the case even when the inputs are restricted to pure Horn 3-CNFs with $O(n^{1+\varepsilon})$ clauses, for some small positive constant $\varepsilon$. Furthermore, we show that even allowing sub-exponential time computation, it is still not possible to obtain constant factor approximations for such problems unless the Exponential Time Hypothesis turns out to be false.