Focused Local Search for Random 3-Satisfiability

TL;DR

This paper investigates focused local search algorithms for random 3-SAT, optimizing parameters to extend their linear-time solving regime closer to the satisfiability threshold, and analyzes their performance and phase transitions.

Contribution

It introduces parameter optimization for focused local search algorithms, extending their effective solving regime near the critical threshold of 3-SAT.

Findings

Focused algorithms can operate in linear time closer to the threshold than previously thought.

Optimal parameters significantly improve the algorithms' performance.

Performance relates to the whitening process and exhibits phase transition behavior.

Abstract

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.