Phase transition in a random NK landscape model

An analysis for the phase transition in a random NK landscape model is given. For the fixed ratio model, NK(<i>n</i>,<i>k</i>,<i>z</i>), Gao and Culberson [17] showed that a random instance generated by NK(<i>n</i>,2,<i>z</i>) with z > z<inf>0</inf> = 27-7/√54 is asymptotically insoluble. Based on empirical results, they conjectured that the phase transition occurs around the value z = z<inf>0</inf>. We prove that an instance generated by NK(<i>n</i>,2,<i>z</i>) with z < z<inf>0</inf> is soluble with positive probability by providing a variant of the unit clause algorithm. Using branching process arguments, we also reprove that an instance generated by NK(<i>n</i>,2,<i>z</i>) with z > z<inf>0</inf> is asymptotically insoluble. The results show the phase transition around z = z<inf>0</inf> for NK(<i>n</i>,2,<i>z</i>). In the course of the analysis, we introduce a generalized random 2-SAT formula, which is of self interest, and show its phase transition phenomenon.

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