  Message-passing (MP) is a powerful tool for finding an approximate solution
in optimization. We generalize it to nonlinear product-sum form, and
numerically show the fast convergence for the minimum feedback vertex set and
the minimum vertex cover known as NP-hard problems. From the linearity of MP in
a logarithmic space, it is derived that an equilibrium solution exists in a
neighborhood of random initial values. These results will give one of the
reason why the convergence is very fast in collective computation based on a
common mathematical background.
