Schilling C., Smaus JG., Wenzelmann F. (2013) A Pretty Complete Combinatorial Algorithm for the Threshold Synthesis Problem. In: Lecroq T., Mouchard L. (eds) Combinatorial Algorithms. IWOCA 2013. Lecture Notes in Computer Science, vol 8288. Springer, Berlin, Heidelberg
A linear pseudo-Boolean constraint (LPB) [1,4,5] is an expression of the form a1ℓ1 + … + am ℓm ≥ d. Here each ℓi is a literal of the form xi or 1 –xi. An LPB can be used to represent a Boolean function; e.g. 2x1 + x2 + x3 ≥ 2 represents the same function as the propositional formula x1 ∨ (x2 ∧ x3).
Functions that can be represented by a single LPB are called threshold functions. The problem of finding the LPB for a threshold function given as disjunctive normal form (DNF) is called threshold synthesis problem. The reference on Boolean functions  formulates the research challenge of recognising threshold functions through an entirely combinatorial procedure. In fact, such a procedure had been proposed in [3,2] and was later reinvented by us . In this paper, we report on an implementation of this procedure for which we have run experiments for up to m = 22. It can solve the biggest problems in a couple of seconds.