Recognition problems for special classes of polynomials in 0–1 variables

This paper investigates the complexity of various recognition problems for pseudo-Boolean functions (i.e., real-valued functions defined on the unit hypercubeBn = {0, 1}n), when such functions are represented as multilinear polynomials in their variables. Determining whether a pseudo-Boolean function (a) is monotonic, or (b) is supermodular, or (c) is threshold, or (d) has a unique local maximum in each face ofBn, or (e) has a unique local maximum inBn, is shown to be NP-hard. A polynomial-time recognition algorithm is presented for unimodular functions, previously introduced by Hansen and Simeone as a class of functions whose maximization overBn is reducible to a network minimum cut problem.

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