Focused Local Search for Random 3-Satisfiability
<|reference_start|>Focused Local Search for Random 3-Satisfiability: 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.<|reference_end|>
@article{seitz2005focused,
    title={Focused Local Search for Random 3-Satisfiability},
    author={Sakari Seitz, Mikko Alava, and Pekka Orponen},
    journal={arXiv preprint arXiv:cond-mat/0501707},
    year={2005},
    doi={10.1088/1742-5468/2005/06/P06006},
    archivePrefix={arXiv},
    eprint={cond-mat/0501707},
    primaryClass={cond-mat.stat-mech cs.CC}
}