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 (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.
Highlights
We have shown that for σ < 0 and large enough |σ|, the phase diagram as a function of α shows the characteristic phenomenology associated with the Random First Order Transition (RFOT) mean field theory of glasses [27, 33,34,35,36]
Our main result is that the jamming transition, which can be seen as the SAT/UNSAT threshold, is always associated with full replica symmetry breaking in the non-convex regime
We have shown that approaching jamming from the SAT phase, one obtains the same critical exponents of the jamming transition of hard spheres in high dimension, reproducing the results of [13]
Summary
The exact description of the glassy phases of high-dimensional sphere systems and the discovery that universal predictions at jamming match finite dimensional observations [1] has renewed the interest in the statistical physics of random constraint satisfaction problems (CSP) with continuous variables. Jamming is the point where the volume of the set of solutions to the problem continuously shrinks to zero, and in its vicinity scaling laws can emerge To this aim, in [14] it was suggested to study the random perceptron problem [15] as a prototype of a CSP with continuous variables, generalized to a region of non-convex optimization. In [19], the avalanches characterizing the glassy phase around jamming have been studied Thanks to these studies, the non-convex perceptron emerges as the simplest model that captures, at the mean field level, all the most important features of the glass and jamming transitions. We present concluding remarks and perspectives for future work
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