Satisfiability thresholds beyond k-XORSAT

TL;DR

This paper investigates the satisfiability thresholds of random systems of equations modulo 3 and over 4-element domains, extending previous results and employing variance-based proof techniques.

Contribution

It extends the understanding of satisfiability thresholds beyond k-XORSAT to systems modulo 3 and over 4-element domains, with new variance analysis methods.

Findings

Threshold occurs where the 2-core has density 1 for k >= 15

Similar threshold results for uniquely extendible constraints over 4 elements

Extension of previous results for equations mod 2 and k=3

Abstract

We consider random systems of equations x_1 + ... + x_k = a; 0 <= a <= 2 which are interpreted as equations modulo 3: We show for k >= 15 that the satisfiability threshold of such systems occurs where the 2-core has density 1: We show a similar result for random uniquely extendible constraints over 4 elements. Our results extend previous results of Dubois/Mandler for equations mod 2 and k = 3 and Connamacher/Molloy for uniquely extendible constraints over a domain of 4 elements with k = 3 arguments. Our proof technique is based on variance calculations, using a technique introduced Dubois/Mandler. However, several additional observations (of independent interest) are necessary.