Finite-size scaling in random $K$-satisfiability problems

Details Finite-size scaling in random $K$-satisfiability problems Finite-size scaling in random $K$-satisfiability problems

2010-05-03

arxiv.org/abs/1005.0251v4

PRE v82, 061109 (2010)

Physics

Condensed Matter

Statistical Mechanics

5 pages, 3 figures (6 eps files), 1 table; published version

Scientific paper

10.1103/PhysRevE.82.061109

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 $2\le K \le 3$.

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