Quadratization of Symmetric Pseudo-Boolean Functions

A pseudo-Boolean function is a real-valued function $f(x)=f(x_1,x_2,\ldots,x_n)$ of $n$ binary variables; that is, a mapping from $\{0,1\}^n$ to $\mathbb{R}$. For a pseudo-Boolean function $f(x)$ on $\{0,1\}^n$, we say that $g(x,y)$ is a quadratization of $f$ if $g(x,y)$ is a quadratic polynomial depending on $x$ and on $m$ auxiliary binary variables $y_1,y_2,\ldots,y_m$ such that $f(x)= \min \{g(x,y) : y \in \{0,1\}^m \}$ for all $x \in \{0,1\}^n$. By means of quadratizations, minimization of $f$ is reduced to minimization (over its extended set of variables) of the quadratic function $g(x,y)$. This is of some practical interest because minimization of quadratic functions has been thoroughly studied for the last few decades, and much progress has been made in solving such problems exactly or heuristically. A related paper \cite{ABCG} initiated a systematic study of the minimum number of auxiliary $y$-variables required in a quadratization of an arbitrary function $f$ (a natural question, since the complexity of minimizing the quadratic function $g(x,y)$ depends, among other factors, on the number of binary variables). In this paper, we determine more precisely the number of auxiliary variables required by quadratizations of symmetric pseudo-Boolean functions $f(x)$, those functions whose value depends only on the Hamming weight of the input $x$ (the number of variables equal to $1$).Comment: 17 page

Quadratic reformulations of nonlinear binary optimization problems

Very large nonlinear unconstrained binary optimization problems arise in a broad array of applications. Several exact or heuristic techniques have proved quite successful for solving many of these problems when the objective function is a quadratic polynomial. However, no similarly efficient methods are available for the higher degree case. Since high degree objectives are becoming increasingly important in certain application areas, such as computer vision, various techniques have been recently developed to reduce the general case to the quadratic one, at the cost of increasing the number of variables. In this paper we initiate a systematic study of these quadratization approaches. We provide tight lower and upper bounds on the number of auxiliary variables needed in the worst-case for general objective functions, for bounded-degree functions, and for a restricted class of quadratizations. Our upper bounds are constructive, thus yielding new quadratization procedures. Finally, we completely characterize all ``minimal'' quadratizations of negative monomials

A polynomial-time algorithm for optimization of quadratic pseudo-boolean functions

We develop a polynomial-time algorithm to minimize pseudo-Boolean functions. The computational complexity is O(n □(15/2)), although very conservative, it is su_cient to prove that this minimization problem is in the class P. A direct application of the algorithm is the 3-SAT problem, which is also guaranteed to be in the class P with a computational complexity of order O(n □(45/2)). The algorithm was implemented in MATLAB and checked by generating one million matrices of arbitrary dimension up to 24 with random entries in the range [-50; 50]. All the experiments were successful

On Symmetric Pseudo-Boolean Functions: Factorization, Kernels and Applications

A symmetric pseudo-Boolean function is a map from Boolean tuples to real numbers which is invariant under input variable interchange. We prove that any such function can be equivalently expressed as a power series or factorized. The kernel of a pseudo-Boolean function is the set of all inputs that cause the function to vanish identically. Any $n$-variable symmetric pseudo-Boolean function $f(x_1, x_2, \dots, x_n)$ has a kernel corresponding to at least one $n$-affine hyperplane, each hyperplane is given by a constraint $\sum_{l=1}^n x_l = \lambda$ for $\lambda\in \mathbb{C}$ constant. We use these results to analyze symmetric pseudo-Boolean functions appearing in the literature of spin glass energy functions (Ising models), quantum information and tensor networks.Comment: 10 page