## Universality of the SAT-UNSAT (jamming) threshold in non-convex continuous constraint satisfaction problems

Silvio Franz, Giorgio Parisi, Maksim Sevelev, Pierfrancesco Urbani, Francesco Zamponi

SciPost Phys. 2, 019 (2017) · published 2 June 2017

- doi: 10.21468/SciPostPhys.2.3.019
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### Abstract

Random constraint satisfaction problems (CSP) have been studied extensively using statistical physics techniques. They provide a benchmark to study average case scenarios instead of the worst case one. The interplay between statistical physics of disordered systems and computer science has brought new light into the realm of computational complexity theory, by introducing the notion of clustering of solutions, related to replica symmetry breaking. However, the class of problems in which clustering has been studied often involve discrete degrees of freedom: standard random CSPs are random K-SAT (aka disordered Ising models) or random coloring problems (aka disordered Potts models). In this work we consider instead problems that involve continuous degrees of freedom. The simplest prototype of these problems is the perceptron. Here we discuss in detail the full phase diagram of the model. In the regions of parameter space where the problem is non-convex, leading to multiple disconnected clusters of solutions, the solution is critical at the SAT/UNSAT threshold and lies in the same universality class of the jamming transition of soft spheres. We show how the critical behavior at the satisfiability threshold emerges, and we compute the critical exponents associated to the approach to the transition from both the SAT and UNSAT phase. We conjecture that there is a large universality class of non-convex continuous CSPs whose SAT-UNSAT threshold is described by the same scaling solution.

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### Ontology / Topics

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^{1 }^{2 }^{3 }^{4 }Silvio Franz, -
^{5 }^{6 }Giorgio Parisi, -
^{1 }^{2 }^{3 }^{4 }Maxime Sevelev, -
^{2 }^{4 }Pierfrancesco Urbani, -
^{2 }^{7 }Francesco Zamponi

^{1}Laboratoire de Physique Théorique de la Matière Condensée, Université Pierre et Marie Curie [LPTMC]^{2}Centre National de la Recherche Scientifique / French National Centre for Scientific Research [CNRS]^{3}Université Paris-Sud / University of Paris-Sud^{4}Université Paris-Saclay / University of Paris-Saclay^{5}Istituto Nazionale di Fisica Nucleare Sezione di Bologna [INFN Bologna]^{6}Sapienza – Università di Roma / Sapienza University of Rome^{7}École Normale Supérieure [ENS]