Very large nonlinear unconstrained binaryoptimization 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 ob...
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Very large nonlinear unconstrained binaryoptimization 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 by introducing additional auxiliary 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.
The problem of minimizing a pseudo-Boolean function, that is, a real-valued function of 0-1 variables, arises in many applications. A quadratization is a reformulation of this nonlinear problem into a quadratic one, o...
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The problem of minimizing a pseudo-Boolean function, that is, a real-valued function of 0-1 variables, arises in many applications. A quadratization is a reformulation of this nonlinear problem into a quadratic one, obtained by introducing a set of auxiliary binary variables. A desirable property for a quadratization is to introduce a small number of auxiliary variables. We present upper and lower bounds on the number of auxiliary variables required to define a quadratization for several classes of specially structured functions, such as functions with many zeros, symmetric, exact k-out-of-n, at least k-out-of-n and parity functions, and monomials with a positive coefficient, also called positive monomials. Most of these bounds are logarithmic in the number of original variables, and we prove that they are best possible for several of the classes under consideration. For positive monomials and for some other symmetric functions, a logarithmic bound represents a significant improvement with respect to the best bounds previously published, which are linear in the number of original variables. Moreover, the case of positive monomials is particularly interesting: indeed, when a pseudo-Boolean function is represented by its unique multilinear polynomial expression, a quadratization can be obtained by separately quadratizing its monomials.
It is proved that any pseudo-Boolean function f can be represented as f(x) equivalent to z + phi(x,(x) over bar), where z is the minimum of f and phi is a polynomial with positive coefficients in the original variable...
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It is proved that any pseudo-Boolean function f can be represented as f(x) equivalent to z + phi(x,(x) over bar), where z is the minimum of f and phi is a polynomial with positive coefficients in the original variables x(i) and in their complements (x) over bar (i). A non-constructivc proof and a constructive one are given. The latter, which is based on a generalization to pseudo-Boolean functions of the well-known Boolean-theoretical operation ofconsensus, provides a new algorithni for the minimization of pseudo-Boolean functions. (c) 2007 Elsevier B.V. All rights reserved.
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