All 4-variable functions can be perfectly quadratized with only 1 auxiliary variable

TL;DR

This paper proves that any 4-variable real-valued function can be exactly transformed into a quadratic function with only one auxiliary variable, significantly reducing complexity and auxiliary variable count compared to previous methods.

Contribution

The authors provide a constructive proof and explicit formulas for perfect quadratizations of all 4-variable functions, enabling more efficient transformations in high-dimensional problems.

Findings

Quadratizations require only 1 auxiliary variable for 4-variable functions.

The new method reduces auxiliary variables from 2N to N in large functions.

Coefficient ranges in quadratic functions are significantly smaller with the new approach.

Abstract

We prove that any function with real-valued coefficients, whose input is 4 binary variables and whose output is a real number, is perfectly equivalent to a quadratic function whose input is 5 binary variables and is minimized over the new variable. Our proof is constructive: we provide quadratizations for all possible 4-variable functions. There exists 4 different classes of 4-variable functions that each have their own 5-variable quadratization formula. Since we provide 'perfect' quadratizations, we can apply these formulas to any 4-variable subset of an n-variable function even if n >> 4. We provide 5 examples of functions that can be quadratized using the result of this work. For each of the 5 examples we compare the best possible quadratization we could construct using previously known methods, to a quadratization that we construct using our new result. In the most extreme example, the quadratization using our new result needs only N auxiliary variables for a 4N-variable degree-4 function, whereas the previous state-of-the-art quadratization requires 2N (double as many) auxiliary variables and therefore we can reduce by the cost of optimizing such a function by a factor of 2^1000 if it were to have 4000 variables before quadratization. In all 5 of our examples, the range of coefficient sizes in our quadratic function is smaller than in the previous state-of-the-art one, and our coefficient range is a factor of 7 times smaller in our 15-term, 5-variable example of a degree-4 function.