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Frozen variables in random boolean constraint satisfaction problems
, 2012
"... We determine the exact freezing threshold, r f, for a family of models of random boolean constraint satisfaction problems, including NAESAT and hypergraph 2colouring, when the constraint size is sufficiently large. If the constraintdensity of a random CSP, F, in our family is greater than r f the ..."
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We determine the exact freezing threshold, r f, for a family of models of random boolean constraint satisfaction problems, including NAESAT and hypergraph 2colouring, when the constraint size is sufficiently large. If the constraintdensity of a random CSP, F, in our family is greater than r f then for almost every solution of F, a linear number of variables are frozen, meaning that their colours cannot be changed by a sequence of alterations in which we change o(n) variables at a time, always switching to another solution. If the constraintdensity is less than r f, then almost every solution has o(n) frozen variables. Freezing is a key part of the clustering phenomenon that is hypothesized by nonrigorous techniques from statistical physics. The understanding of clustering has led to the development of advanced heuristics such as Survey Propogation. It has been suggested that the freezing threshold is a precise algorithmic barrier: There is reason to believe that for densities below r f the random CSPs can be solved using very simple algorithms, while for densities above r f one requires more sophisticated techniques in order to deal with frozen clusters. 0 1