TY - GEN

T1 - The Number of Solutions for Random Regular NAE-SAT

AU - Sly, Allan

AU - Sun, Nike

AU - Zhang, Yumeng

PY - 2016/12/14

Y1 - 2016/12/14

N2 - Recent work has made substantial progress in understanding the transitions of random constraint satisfaction problems (CSPs). In particular, for several of these models, the exact satisfiability threshold has been rigorously determined, confirming predictions from the statistical physics literature. Here we revisit one of these models, random regular NAE-SAT: knowing the satisfiability threshold, it is natural to study, in the satisfiable regime, the number of solutions in a typical instance. We prove here that these solutions have a well-defined free energy (limiting exponential growth rate), with explicit value matching the one-step replica symmetry breaking prediction. The proof develops new techniques for analyzing a certain 'survey propagation model' associated to this problem. We believe that these methods may be applicable in a wide class of related problems.

AB - Recent work has made substantial progress in understanding the transitions of random constraint satisfaction problems (CSPs). In particular, for several of these models, the exact satisfiability threshold has been rigorously determined, confirming predictions from the statistical physics literature. Here we revisit one of these models, random regular NAE-SAT: knowing the satisfiability threshold, it is natural to study, in the satisfiable regime, the number of solutions in a typical instance. We prove here that these solutions have a well-defined free energy (limiting exponential growth rate), with explicit value matching the one-step replica symmetry breaking prediction. The proof develops new techniques for analyzing a certain 'survey propagation model' associated to this problem. We believe that these methods may be applicable in a wide class of related problems.

KW - Condensation transition

KW - Free energy

KW - One-step replica symmetry breaking

KW - Random constraint satisfaction problem

KW - Replica symmetry

KW - Satisfiability threshold

UR - http://www.scopus.com/inward/record.url?scp=85009381609&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85009381609&partnerID=8YFLogxK

U2 - 10.1109/FOCS.2016.82

DO - 10.1109/FOCS.2016.82

M3 - Conference contribution

AN - SCOPUS:85009381609

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 724

EP - 731

BT - Proceedings - 57th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2016

PB - IEEE Computer Society

T2 - 57th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2016

Y2 - 9 October 2016 through 11 October 2016

ER -