John Ardelius et al J. Stat. Mech. (2007) P10012 doi:10.1088/1742-5468/2007/10/P10012
John Ardelius1, Erik Aurell2 and Supriya Krishnamurthy3
Show affiliationsWe study numerically the solution space structure of random 3-SAT problems close to the SAT/UNSAT transition. This is done by considering chains of satisfiability problems, where clauses are added sequentially to a problem instance. Using the overlap measure of similarity between different solutions found on the same problem instance, we examine geometrical changes as a function of α. In each chain, the overlap distribution is first smooth, but then develops a tiered structure, indicating that the solutions are found in well separated clusters. On chains of not too large instances, all remaining solutions are eventually observed to be found in only one small cluster before vanishing. This condensation transition point is estimated by finite size scaling to be αc = 4.26 with an apparent critical exponent of about 1.7. The average overlap value is also observed to increase with α up to the transition, indicating a reduction in solutions space size, in accordance with theoretical predictions. The solutions are generated by a local heuristic, ASAT, and compared to those found by the Survey Propagation algorithm up to αc.
E-print Number: cond-mat/0702672
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Refers: to
05.40.Fb Random walks and Levy flights
02.60.Gf Algorithms for functional approximation
05.70.Jk Critical point phenomena
62H30 Classification and discrimination; cluster analysis (See also 68T10)
82B41 Random walks, random surfaces, lattice animals, etc. (See also 60G50, 82C41)
Issue 10 (October 2007)
Received 21 June 2007, accepted for publication 4 October 2007
Published 24 October 2007
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