  We provide a theoretical study of the quantum adiabatic evolution algorithm
with different evolution paths proposed in [E. Farhi, et al.,
arXiv:quant-ph/0208135]. The algorithm is applied to a random binary
optimization problem (a version of the 3-Satisfiability problem) where the
n-bit cost function is symmetric with respect to the permutation of individual
bits. The evolution paths are produced, using the generic control Hamiltonians
H(\tau) that preserve the bit symmetry of the underlying optimization problem.
In the case where the ground state of H(0) coincides with the totally-symmetric
state of an n-qubit system the algorithm dynamics is completely described in
terms of the motion of a spin-n/2.
  We show that different control Hamiltonians can be parameterized by a set of
independent parameters that are expansion coefficients of H(\tau) in a certain
universal set of operators. Only one of these operators can be responsible for
avoiding the tunnelling in the spin-n/2 system during the quantum adiabatic
algorithm. We show that it is possible to select a coefficient for this
operator that guarantees a polynomial complexity of the algorithm for all
problem instances. We show that a successful evolution path of the algorithm
always corresponds to the trajectory of a classical spin-n/2 and provide a
complete characterization of such paths.
