Trajectories in Phase Diagrams, Growth Processes, and Computational Complexity: How Search Algorithms Solve the 3-Satisfiability Problem

Abstract

Trajectories in phase diagrams, growth processes and computational complexity:how search algorithms solve the 3-Satisfiability problem. Most decision and optimization problems encountered in practice fall into one of two categories with respect to any particular solving method or algorithm: either the problem is solved quickly (easy) or else demands an impractically long computational effort (hard). Recent investigations on model classes of problems have shown that some global parameters, such as the ratio between the constraints to be satisfied and the adjustable variables, are good predictors of problem hardness and, moreover, have an effect analogous to thermodynamical parameters, e.g. temperature, in predicting phases in condensed matter physics [Monasson et al., Nature 400 (1999) 133-137]. Here we show that changes in the values of such parameters can be tracked during a run of the algorithm defining a trajectory through the parameter space. Focusing on 3-Satisfiability, a recognized representative of hard problems, we analyze trajectories generated by search algorithms using growth processes statistical physics. These trajectories can cross well defined phases, corresponding to domains of easy or hard instances, and allow to successfully predict the times of resolution.

Phase transitions in the coloring of random graphs

“…It is much more convenient to look at the ensemble of random regular graphs where a factorized solution P[P (ψ)] = δ(P (ψ) − P 0 (ψ)) must exists. Then we are left with only one functional equation (20). Before discussing the zero temperature limit, we would like to point out that there exists another very important case in which eq.…”

“…(20). Solving this equation for a general ensemble of random graphs and a general parameter m is a numerically quite tedious problem.…”

“…The notation POP comes from "population dynamics" which refers to the numerical method we use to solve eq. (20). The probability distribution P (ψ) can be represented numerically by a set of fields taken from P (ψ), and then the probability measure P (ψ)dψ becomes uniform sampling from this set, for more details see appendix D.…”

“…For simplicity we will write them only for the 3-coloring. We consider the 1RSB cavity equation (20) for m → 0, then the reweighting factor (Z i→j 0 ) m is equal to zero when the arriving hard fields are contradictory, and equal to one otherwise. The update of probability η s that a field is frozen in direction s is then written from eq.…”

“…This onset of computational hardness cannot be explained only by the simple fact that near to the colorable threshold the number of proper colorings is small [18]. Some progress in the theoretical understanding has been done by the analysis of search algorithms [19,20]. For the coloring problem, it was proven [21] that a simple algorithm q-colors almost surely in linear time random graphs of average connectivity c ≤ q log q − 3q/2 for all q ≥ 3 (see [21] for references on previous works).…”