A Pretty Complete Combinatorial Algorithm for the Threshold Synthesis Problem

Abstract : A linear pseudo-Boolean constraint (LPB) is an expression of the form a_1 l_1+...+a_m l_m>= d$, where each l_i is a literal (it assumes the value 1 or 0 depending on whether a propositional variable x_i is true or false) and a_1,...,a_m,d are natural numbers. An LPB represents a Boolean function, and those Boolean functions that can be represented by exactly one LPB are called threshold functions. The problem of finding an LPB representation of a Boolean function if possible is called threshold recognition problem or threshold synthesis problem. The problem has an O(m^7t^5) algorithm using linear programming, where m is the dimension and t the number of clauses in the DNF input. There is also an entirely combinatorial procedure, which works by decomposing the DNF and "counting" the variable occurrences in it. We have implemented both algorithms and report here on the experiments.
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Christian Schilling, Jan-Georg Smaus, Fabian Wenzelmann. A Pretty Complete Combinatorial Algorithm for the Threshold Synthesis Problem. International Workshop on Combinatorial Algorithms (IWOCA 2013), Jul 2013, Rouen, France. pp. 458-462. ⟨hal-01671327⟩

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