  In this paper we analyze the performance of the Quantum Adiabatic Evolution
algorithm on a variant of Satisfiability problem for an ensemble of random
graphs parametrized by the ratio of clauses to variables, $\gamma=M/N$. We
introduce a set of macroscopic parameters (landscapes) and put forward an
ansatz of universality for random bit flips. We then formulate the problem of
finding the smallest eigenvalue and the excitation gap as a statistical
mechanics problem. We use the so-called annealing approximation with a
refinement that a finite set of macroscopic variables (versus only energy) is
used, and are able to show the existence of a dynamic threshold
$\gamma=\gamma_d$ starting with some value of K -- the number of variables in
each clause. Beyond dynamic threshold, the algorithm should take exponentially
long time to find a solution. We compare the results for extended and
simplified sets of landscapes and provide numerical evidence in support of our
universality ansatz. We have been able to map the ensemble of random graphs
onto another ensemble with fluctuations significantly reduced. This enabled us
to obtain tight upper bounds on satisfiability transition and to recompute the
dynamical transition using the extended set of landscapes.
