IWOCA 2013: Combinatorial Algorithms pp 458-462

# A Pretty Complete Combinatorial Algorithm for the Threshold Synthesis Problem

• Christian Schilling
• Jan-Georg Smaus
• Fabian Wenzelmann
Conference paper

DOI: 10.1007/978-3-642-45278-9_43

Volume 8288 of the book series Lecture Notes in Computer Science (LNCS)
Cite this paper as:
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

## Abstract

A linear pseudo-Boolean constraint (LPB) [1,4,5] is an expression of the form a11 + … + amm ≥ 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 [4] 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 [7]. 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.