  A local search algorithm solving an NP-complete optimisation problem can be
viewed as a stochastic process moving in an 'energy landscape' towards
eventually finding an optimal solution. For the random 3-satisfiability
problem, the heuristic of focusing the local moves on the presently
unsatisfiedclauses is known to be very effective: the time to solution has been
observed to grow only linearly in the number of variables, for a given
clauses-to-variables ratio $\alpha$ sufficiently far below the critical
satisfiability threshold $\alpha_c \approx 4.27$. We present numerical results
on the behaviour of three focused local search algorithms for this problem,
considering in particular the characteristics of a focused variant of the
simple Metropolis dynamics. We estimate the optimal value for the
``temperature'' parameter $\eta$ for this algorithm, such that its linear-time
regime extends as close to $\alpha_c$ as possible. Similar parameter
optimisation is performed also for the well-known WalkSAT algorithm and for the
less studied, but very well performing Focused Record-to-Record Travel method.
We observe that with an appropriate choice of parameters, the linear time
regime for each of these algorithms seems to extend well into ratios $\alpha >
4.2$ -- much further than has so far been generally assumed. We discuss the
statistics of solution times for the algorithms, relate their performance to
the process of ``whitening'', and present some conjectures on the shape of
their computational phase diagrams.
