We determine the complexity of an optimization problem related to information theory. Taking a conjunctive propositional formula over some finite set of Boolean relations as input, we seek a satisfying assignment of the formula having minimal Hamming distance to a given assignment that is not required to be a model (NearestSolution, NSol). We obtain a complete classification with respect to the relations admitted in the formula. For two classes of constraint languages we present polynomial time algorithms; otherwise, we prove hardness or completeness concerning the classes APX, poly-APX, NPO, or equivalence to well-known hard optimization problems.
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