Implementations of two Algorithms for the Threshold Synthesis Problem

TL;DR

This paper compares two algorithms for recognizing threshold functions, implementing and analyzing their effectiveness, revealing that Smaus's combinatorial method is incomplete, which impacts its practical applicability.

Contribution

The paper provides the first implementation and comparison of the linear programming and combinatorial algorithms for threshold recognition, highlighting limitations of Smaus's method.

Findings

Smaus's combinatorial algorithm is incomplete.

The LP-based algorithm has a known polynomial-time complexity.

Practical implementation reveals limitations in the combinatorial approach.

Abstract

A linear pseudo-Boolean constraint (LPB) is an expression of the form $a_1 \cdot \ell_1 + \dots + a_m \cdot \ell_m \geq d$, where each $\ell_i$ is a literal (it assumes the value 1 or 0 depending on whether a propositional variable $x_i$ is true or false) and $a_1, \dots, a_m, d$ 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 $O(m^7 t^5)$ algorithm using linear programming, where $m$ is the dimension and $t$ the number of terms in the DNF input. It has been an open question whether one can recognise threshold functions through an entirely combinatorial procedure. Smaus has developed such a procedure for doing this, which works by decomposing the DNF and "counting" the variable occurrences in it. We have implemented both algorithms as a thesis project. We report here on this experience. The most important insight was that the algorithm by Smaus is, unfortunately, incomplete.