Quadratizations of symmetric pseudo-Boolean functions: sub-linear bounds on the number of auxiliary variables

The problem of minimizing a pseudo-Boolean function with no additional constraints arises in a variety of applications. A quadratization is a quadratic reformulation of the nonlinear problem obtained by introducing a set of auxiliary binary variables which can be optimized using quadratic optimization techniques. Using the well-known result that a pseudoBoolean function can be uniquely expressed as a multilinear polynomial, any pseudo-Boolean function can be quadratized by providing a quadratization for negative monomials and one for positive monomials. A desirable property for a quadratization is to introduce small number of auxiliary variables. For the case of negative monomials there exists a quadratization using a single auxiliary variable which is, in addition, submodular. However, much less is known for positive monomials. The best lower bound on the number of variables required by a quadratization in the literature is b n−1 2 c. We present here new quadratizations of the positive monomial that significantly improve this lower bound decreasing it to a logarithmic bound. This lower bound is derived from a more general result, stating that a quadratization with a logarithmic number of variables can be defined for exact kout-of-n functions, which is a more general class of symmetric functions. Moreover, for exact k-out-of-n functions we also prove that a logarithmic number of variables is necessary when requiring certain symmetry conditions to the quadratization. Finally, we provide quadratizations for general symmetric functions using O( √ n) auxiliary variables. This upper bound nicely matches a lower bound of Ω( √ n) variables that was recently introduced. Introduction A pseudo-Boolean function is a mapping f : {0, 1}n → R that assigns a real value to each tuple of n binary variables (x1, . . . , xn). Pseudo-Boolean functions have been extensively used and studied during the last century and especially in the last 50 years, given that they model problems in a wide range of areas such as reliability theory, computer science, statistics, economics, finance, operations research, management science, discrete mathematics, or computer vision (see (Boros and Hammer 2002) and (Crama and Hammer 2011) ∗endre.boros@rutgers.edu †yves.crama@ulg.ac. be ‡elisabeth.rodriguezheck@ulg.ac. be for a list of applications and references). In most of these applications f has to be optimized, therefore we are interested in the problem min x∈{0,1}n f (x), (1) which is NP-hard even when f is a quadratic function. Several techniques have been proposed to solve problem (1), such as enumerative methods, algebraic methods, linear reformulations and quadratic reformulations, which are then solved using a linear or quadratic solver, respectively. It is not clear whether one of the previous techniques is generally better than the others. In fact, the performance of the different approaches seems to depend on the underlying structure of the problem, among other factors. In this paper, we focus on quadratic reformulations of problem (1), which are also called quadratizations. Interestingly, much progress in the understanding of quadratizations has been made in the field of computer vision, where these type of techniques perform especially well for problems such as image restoration. A systematic study of quadratizations and its properties has been initiated in (Anthony et al. 2017), where a quadratization is defined as follows. Definition 1. Given 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 variables y1, . , ym, such that f (x) = min y∈{0,1}m g(x, y) ∀x ∈ {0, 1}n. (2) It is clear that given a pseudo-Boolean function f and a quadratization g, minimizing f over x ∈ {0, 1}n is equivalent to minimizing g over the (x, y) ∈ {0, 1}n+m. Moreover, it is known that every pseudo-Boolean function f admits a quadratization (Rosenberg 1975). Not all quadratizations will perform equally well when solving the resulting quadratic problem. A desirable property of a quadratization is to have a small set of auxiliary variables, so that the size of the reformulation does not increase too much with respect to the size of the original problem. Another interesting property is submodularity. Of course, if the original pseudo-Boolean function f is not submodular, then it does not admit a submodular quadratization. However, even if a quadratization is not submodular, a small number of positive quadratic terms might have a positive impact in resolution performance. Literature review Given a pseudo-Boolean function f , there are many different methods to define a quadratization. In particular, termwise quadratizations have attracted much interest in the literature. This type of procedure uses the well-known result that a pseudo-Boolean function f can be represented by a unique multilinear polynomial (Hammer, Rosenberg, and Rudeanu 1963), (Hammer and Rudeanu 1968), i.e., f (x1, . . . , xn) = ∑