Learning from Survey Propagation: a Neural Network for MAX-E-33-SAT

Many natural optimization problems are NP-hard, which implies that they are probably hard to solve exactly in the worst-case. However, it suffices to get reasonably good solutions for all (or even most) instances in practice. This paper presents a new algorithm for computing approximate solutions in ${\Theta(N})$ for the Maximum Exact 3-Satisfiability (MAX-E-$3$-SAT) problem by using deep learning methodology. This methodology allows us to create a learning algorithm able to fix Boolean variables by using local information obtained by the Survey Propagation algorithm. By performing an accurate analysis, on random CNF instances of the MAX-E-$3$-SAT with several Boolean variables, we show that this new algorithm, avoiding any decimation strategy, can build assignments better than a random one, even if the convergence of the messages is not found. Although this algorithm is not competitive with state-of-the-art Maximum Satisfiability (MAX-SAT) solvers, it can solve substantially larger and more complicated problems than it ever saw during training

The backtracking survey propagation algorithm for solving random K-SAT problems

Discrete combinatorial optimization has a central role in many scientific disciplines, however, for hard problems we lack linear time algorithms that would allow us to solve very large instances. Moreover, it is still unclear what are the key features that make a discrete combinatorial optimization problem hard to solve. Here we study random K-satisfiability problems with $K=3,4$, which are known to be very hard close to the SAT-UNSAT threshold, where problems stop having solutions. We show that the backtracking survey propagation algorithm, in a time practically linear in the problem size, is able to find solutions very close to the threshold, in a region unreachable by any other algorithm. All solutions found have no frozen variables, thus supporting the conjecture that only unfrozen solutions can be found in linear time, and that a problem becomes impossible to solve in linear time when all solutions contain frozen variables.Comment: 11 pages, 10 figures. v2: data largely improved and manuscript rewritte

Waves and vortices in the inverse cascade regime of stratified turbulence with or without rotation

We study the partition of energy between waves and vortices in stratified turbulence, with or without rotation, for a variety of parameters, focusing on the behavior of the waves and vortices in the inverse cascade of energy towards the large scales. To this end, we use direct numerical simulations in a cubic box at a Reynolds number Re=1000, with the ratio between the Brunt-V\"ais\"al\"a frequency N and the inertial frequency f varying from 1/4 to 20, together with a purely stratified run. The Froude number, measuring the strength of the stratification, varies within the range 0.02 < Fr < 0.32. We find that the inverse cascade is dominated by the slow quasi-geostrophic modes. Their energy spectra and fluxes exhibit characteristics of an inverse cascade, even though their energy is not conserved. Surprisingly, the slow vortices still dominate when the ratio N/f increases, also in the stratified case, although less and less so. However, when N/f increases, the inverse cascade of the slow modes becomes weaker and weaker, and it vanishes in the purely stratified case. We discuss how the disappearance of the inverse cascade of energy with increasing N/f can be interpreted in terms of the waves and vortices, and identify three major effects that can explain this transition based on inviscid invariants arguments

On the emergence of helicity in rotating stratified turbulence

We perform numerical simulations of decaying rotating stratified turbulence and show, in the Boussinesq framework, that helicity (velocity-vorticity correlation), as observed in super-cell storms and hurricanes, is spontaneously created due to an interplay between buoyancy and rotation common to large-scale atmospheric and oceanic flows. Helicity emerges from the joint action of eddies and of inertia-gravity waves (with inertia and gravity with respective associated frequencies $f$ and $N$), and it occurs when the waves are sufficiently strong. For $N/f < 3$ the amount of helicity produced is correctly predicted by a quasi-linear balance equation. Outside this regime, and up to the highest Reynolds number obtained in this study, namely $Re\approx 10000$, helicity production is found to be persistent for $N/f$ as large as $\approx 17$, and for $ReFr^2$ and $ReRo^2$ respectively as large as $\approx 100$ and $\approx 24000$.Comment: 10 pages, 5 figure

Servitización y mecanismos de autorrefuerzo territorial: un nuevo enfoque a la competitividad regional

The present paper discusses a theoretical model to explain the link between servitization and territorial competitiveness based on the situation in Italy. A key assumption of the model is that once the link between manufacturing and KIBS is established within a TES, there is a positive feedback between the increasing productivity (competitiveness) and the link between firms and KIBS, which becomes stronger and stronger triggering a self-reinforcing dynamic. This means that every evolutionary step of the system influences the next and thus the evolution of the entire system, so generating path dependence. Such a system has a high number of asymptotic states, and the initial state (time zero), unforeseen shocks, or other kinds of fluctuations, can lead the system into any of the different domains of the asymptotic states (1). In other words, both the theoretical assumptions and the empirical model outlined in this paper demonstrate that when a functional relationship between manufacturing and services is established (servitization), economic performance is positive or very positive.El presente artículo analiza un modelo teórico para explicar el vínculo entre servitización y competitividad territorial basado en estudio empírico de tal relación en Italia. Un supuesto clave es que, una vez que se establece el vínculo entre la manufactura y KIBS dentro de un TES, hay una retroalimentación positiva entre el aumento de la productividad (competitividad) y el vínculo entre las empresas productoras y los KIBS se genera una dinámica de retroalimentación positiva que conduce a un aumento de la productividad (competitividad). Esto implica que cada paso evolutivo del sistema influye en el siguiente y, por lo tanto, en la evolución de todo el sistema, se generan interdependencias (generando así dependencia del camino). Tal sistema tiene un alto número de estados asintóticos, y durante el estado inicial (tiempo cero), choques imprevistos u otros tipos de fluctuaciones, pueden llevar al sistema a cualquiera de los diferentes dominios de los estados asintóticos (1). En otras palabras, tanto los supuestos teóricos como el modelo empírico esbozado en este trabajo demuestran que cuando se establece una relación funcional entre manufactura y servicios (servitización), el desempeño económico es positivo o muy positivo