On quadratization of pseudo-Boolean functions

Set functions, i.e., real mappings form the family of subsets of a finite set to the reals are known and widely used in discrete mathematics for almost a century, and in particular in the last 50 years. If we replace a finite set with its characteristic vector, then the same set function can be interpreted as a mapping from the set of binary vectors to the reals. Such mappings are called pseudo-Boolean functions and were introduced in the works of Peter L. Hammer in the 1960s, see the seminal book (Hammer and Rudeanu 1968). PseudoBoolean functions are different from set functions, only in the sense that their algebraic representation, a multilinear polynomial expression, is usually assumed to be available as an input representation:

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