The Binary Linearization Complexity of Pseudo-Boolean Functions
Matthias Walter
TL;DR
This work introduces the concept of linearization complexity for pseudo-Boolean functions, defined via binary auxiliary functions, and studies how many such functions are needed to exactly represent a given $f:\{0,1\}^n\to\mathbb{R}$. It proves that for random polynomials, small linearizations are almost surely impossible, and it characterizes finiteness and computation of $\mathrm{lc}_{\mathcal{G}}(f)$ for various function families, including monomials and arbitrary Boolean functions. The authors then apply two linearization schemes to the low autocorrelation binary sequences problem, deriving IP formulations and evaluating them computationally; a value-indicator-based approach (VIQ) shows promise in reducing model size and improving LP bounds, though it does not yet outperform the best specialized methods. The paper closes with a set of open questions about additional function families, stronger formulations, bounding techniques, and approximation strategies, pointing to rich directions for future research.
Abstract
We consider the problem of linearizing a pseudo-Boolean function $f : \{0,1\}^n \to \mathbb{R}$ by means of $k$ Boolean functions. Such a linearization yields an integer linear programming formulation with only $k$ auxiliary variables. This motivates the definition of the linarization complexity of $f$ as the minimum such $k$. Our theoretical contributions are the proof that random polynomials almost surely have a high linearization complexity and characterizations of its value in case we do or do not restrict the set of admissible Boolean functions. The practical relevance is shown by devising and evaluating integer linear programming models of two such linearizations for the low auto-correlation binary sequences problem. Still, many problems around this new concept remain open.