  Using a simple, annealed model, some of the key features of the recently
introduced extremal optimization heuristic are demonstrated. In particular, it
is shown that the dynamics of local search possesses a generic critical point
under the variation of its sole parameter, separating phases of too greedy
(non-ergodic, jammed) and too random (ergodic) exploration. Comparison of
various local search methods within this model suggests that the existence of
the critical point is essential for the optimal performance of the heuristic.
