  We provide a comprehensive view of various phase transitions in random
$K$-satisfiability problems solved by stochastic-local-search algorithms. In
particular, we focus on the finite-size scaling (FSS) exponent, which is
mathematically important and practically useful in analyzing finite systems.
Using the FSS theory of nonequilibrium absorbing phase transitions, we show
that the density of unsatisfied clauses clearly indicates the transition from
the solvable (absorbing) phase to the unsolvable (active) phase as varying the
noise parameter and the density of constraints. Based on the solution
clustering (percolation-type) argument, we conjecture two possible values of
the FSS exponent, which are confirmed reasonably well in numerical simulations
for $2\le K \le 3$.
