Optimal quadratic reformulations of fourth degree Pseudo-Boolean functions

Abstract

Pseudo-Boolean functions (PBF) are closed algebraic representations of set functions that are closely related to nonlinear binary optimizations and have numerous applications. Algorithms for PBF of degree two (quadratic) are NP-Hard and third and fourth degree functions are increasingly difficult to solve. However, the higher degree terms can be reformulated to a lower degree by adding variables and corresponding penalty constraints. These additional constraints can then be transformed to the objective function via penalties to create Quadratic Unconstrained Binary Optimization problems for which there are many solution techniques, such as tabu search and quantum annealing. Shortcomings of reformulation are the possibility of large numbers of auxiliary variables and constraints along with large penalty terms. In this paper, we address these shortcomings by presenting a preprocessing approach for fourth degree pseudo-Boolean polynomials based on an exact integer programming model that minimizes the number of auxiliary variables and penalty magnitude. Experimental results compare worst case, naive, greedy and minimal substitution methods and illustrate the efficacy of minimizing substitutions and penalty magnitude.